Introduction
We prove Sogge’s local smoothing conjecture for the Euclidean wave equation in every spatial dimension n ≥ 2 n \ge2 n ≥ 2 L p L^p L p [72 Section 4, Equation (4.5)] .
Write U ( T ) = e − i T ∣ D ∣ U(T)=e^{-iT|D|} U ( T ) = e − i T ∣ D ∣ ∣ D ∣ = − Δ |D|=\sqrt{-\Delta} ∣ D ∣ = − Δ I = [ 1 , 2 ] I=[1,2] I = [ 1 , 2 ] J = ( 1 − Δ ) 1 / 2 J=(1-\Delta)^{1/2} J = ( 1 − Δ ) 1/2
s + ( p ) = max { 0 , n − 1 2 − n p } . s_+(p)=\max\left\{0,\frac{n-1}{2}-\frac{n}{p}\right\}. s + ( p ) = max { 0 , 2 n − 1 − p n } . Theorem 1.1 (Euclidean local smoothing). For n ≥ 2 n \ge2 n ≥ 2 2 ≤ p < ∞ 2 \le p < \infty 2 ≤ p < ∞ ε > 0 \varepsilon> 0 ε > 0
∥ U ( T ) f ∥ L p ( R n × I ) ≤ C n , p , ε ∥ J s + ( p ) + ε f ∥ L p ( R n ) . (1) \lVert U(T)f\rVert_{L^p(\mathbb{R}^n\times I)} \le C_{n,p,\varepsilon}\lVert J^{s_+(p)+\varepsilon}f\rVert_{L^p(\mathbb{R}^n)}. \tag*{(1)} ∥ U ( T ) f ∥ L p ( R n × I ) ≤ C n , p , ε ∥ J s + ( p ) + ε f ∥ L p ( R n ) . (1) At p = 2 p=2 p = 2 ε = 0 \varepsilon=0 ε = 0
At a fixed nonzero time the sharp Sobolev order is ( n − 1 ) ( 1 / 2 − 1 / p ) (n-1)(1/2-1/p) ( n − 1 ) ( 1/2 − 1/ p ) [54 , 65 , 68 ] . Thus for p ≥ p c = 2 n / ( n − 1 ) p \ge p_c=2n/(n-1) p ≥ p c = 2 n / ( n − 1 ) 1 / p 1/p 1/ p 2 < p < p c 2<p<p_c 2 < p < p c s + ( p ) s_+(p) s + ( p ) Proposition 8.1 ). We state the wave estimates for complex Schwartz data; density gives the indicated extensions, and conjugation gives the other sign of the half wave. Unqualified L p L^p L p
Consequences for Fourier analysis and geometry
Bochner–Riesz means and Stein’s square function. The Bochner–Riesz means B R α = ( 1 − ∣ D ∣ 2 / R 2 ) + α B_{R}^{\alpha} = (1-|D|^{2}/R^{2})_{+}^{\alpha} B R α = ( 1 − ∣ D ∣ 2 / R 2 ) + α L p L^{p} L p 1 ≤ p ≤ ∞ 1 \leq p \leq\infty 1 ≤ p ≤ ∞ α > max { 0 , n ∣ 1 / p − 1 / 2 ∣ − 1 / 2 } \alpha> \max\{0,n|1/p-1/2|-1/2\} α > max { 0 , n ∣1/ p − 1/2∣ − 1/2 } 2 ≤ p < ∞ 2 \leq p < \infty 2 ≤ p < ∞ L p L^{p} L p [73 ] . Keeping this variation inside a square integral gives its conjectured bounds for 2 ≤ p < ∞ 2 \leq p < \infty 2 ≤ p < ∞ α > 1 / 2 + s + ( p ) \alpha> 1/2+s_{+}(p) α > 1/2 + s + ( p ) [72 Equations (4.5)–(4.6)] ; for the reductions at fixed and variable radii see [8 Propositions 39 and 40] .
Fourier restriction and Schrödinger waves. Give an incoming spherical wave an amplitude that varies with direction. Near the focus, rescaling recovers the Fourier extension of this amplitude from the sphere. We obtain the full interior range of global Fourier extension estimates L a → L b L^{a} \to L^{b} L a → L b 1 < a ≤ ∞ 1 < a \leq\infty 1 < a ≤ ∞ b < ∞ b < \infty b < ∞
b > 2 n n − 1 , b > n + 1 n − 1 a ′ , 1 a + 1 a ′ = 1. b > \frac{2n}{n-1}, \qquad b > \frac{n+1}{n-1}a', \qquad\frac{1}{a}+\frac{1}{a'}=1. b > n − 1 2 n , b > n − 1 n + 1 a ′ , a 1 + a ′ 1 = 1. Slicing gives the same bounds for the truncated cone in R n + 1 \mathbb{R}^{n+1} R n + 1 n − 1 n-1 n − 1
Kakeya and Nikodym sets. Random signs on disjoint frequency caps turn the square of the extension into a bound for overlaps of tubes. This proves the Kakeya maximal conjecture: the operator that takes f f f ∣ f ∣ |f| ∣ f ∣ δ \delta δ O ϵ ( δ − ϵ ) O_{\epsilon}(\delta^{-\epsilon}) O ϵ ( δ − ϵ ) L n ( R n ) L^{n}(\mathbb{R}^{n}) L n ( R n ) L n ( S n − 1 ) L^{n}(S^{n-1}) L n ( S n − 1 ) ϵ > 0 \epsilon> 0 ϵ > 0 R n \mathbb{R}^{n} R n n n n
Development of the local smoothing problem
The question grew out of Bourgain’s theorem on maximal averages over circles [12 ] . At high frequency the Fourier transform of a circle of radius T T T T ∣ ξ ∣ T|\xi| T ∣ ξ ∣ − T ∣ ξ ∣ -T|\xi| − T ∣ ξ ∣ p > 2 p > 2 p > 2 [72 Theorem 4.1] . Mockenhaupt, Seeger and Sogge gave a proof of the circular maximal theorem using angular square functions [55 ] , and developed local smoothing for Fourier integral operators under a curvature condition on the associated family of cones [56 ] .
A persistent difficulty is to put waves moving in different directions back together. Wolff decomposed a neighborhood of the cone into thin angular sectors and controlled their sum by induction on scales. This gave every gain below 1 / p 1/p 1/ p p > 74 p > 74 p > 74 [80 ] ; Łaba and Wolff extended the argument to higher dimensions [49 ] . Bourgain and Demeter’s decoupling theorem gives this gain for p ≥ 2 ( n + 1 ) / ( n − 1 ) p \geq2(n+1)/(n-1) p ≥ 2 ( n + 1 ) / ( n − 1 ) [14 Theorem 1.2] . A different argument using radial Fourier multipliers attained the exact gain 1 / p 1/p 1/ p n ≥ 4 n \geq4 n ≥ 4 p > 2 ( n − 1 ) / ( n − 3 ) p > 2(n-1)/(n-3) p > 2 ( n − 1 ) / ( n − 3 ) [37 Theorem 1.2] . In the plane, Guth, Wang and Zhang reached the critical exponent p = 4 p=4 p = 4 L 4 L^{4} L 4 [35 Theorems 1.1 and 1.2] . For the canonical sectors of a truncated cone, Cushman, Demeter and Wu give the sharp power 1 / 4 1/4 1/4 [20 Theorem 1.4] . To obtain a bound in terms of ∥ f ∥ 4 \|f\|_4 ∥ f ∥ 4
In higher dimensions the distinction between a partial gain at p c p_c p c p c p_c p c n = p = 3 n=p=3 n = p = 3 2 / 9 2/9 2/9 1 / 3 1/3 1/3 [29 Corollary 1.3] . Gan, He, Li and Wu proved the full gain up to an arbitrarily small loss for p ≥ 10 / 3 p\ge10/3 p ≥ 10/3 R 3 \mathbb{R}^3 R 3 [28 Theorem 1.3] ; see also Wang’s study of wave packet density [78 ] . Interpolating the p = 10 / 3 p=10/3 p = 10/3 L 2 L^2 L 2 1 / 4 1/4 1/4 p = 3 p=3 p = 3 p ≥ 10 / 3 p\ge10/3 p ≥ 10/3 1 / p 1/p 1/ p [53 ] , whereas the Euclidean exponent is 3 3 3
An estimate on one frequency annulus
The proof is built around a single estimate at p = p c p=p_c p = p c Theorem 1.1 allows. Our Fourier convention is f ^ ( ξ ) = ∫ R n e − i X ⋅ ξ f ( X ) d X \widehat{f}(\xi)=\int_{\mathbb{R}^n}e^{-iX\cdot\xi}f(X)\,dX f ( ξ ) = ∫ R n e − i X ⋅ ξ f ( X ) d X U ( T ) U(T) U ( T ) X ⋅ ξ − T ∣ ξ ∣ X\cdot\xi-T|\xi| X ⋅ ξ − T ∣ ξ ∣ ( 2 π ) − n (2\pi)^{-n} ( 2 π ) − n
Theorem 1.2 (Critical annular estimate). Let n ≥ 2 n\ge2 n ≥ 2 p = p c p=p_c p = p c φ ∈ C c ∞ ( R n ∖ { 0 } ) \varphi\in C^\infty_c(\mathbb{R}^n\setminus\{0\}) φ ∈ C c ∞ ( R n ∖ { 0 }) N ≥ 2 N\ge2 N ≥ 2 P N = φ ( D / N ) P_N=\varphi(D/N) P N = φ ( D / N )
∥ U ( T ) P N f ∥ L p ( R n × I ) ≤ C n , φ [ log ( 2 + N ) ] 2 / p ∥ f ∥ L p ( R n ) . (2) \|U(T)P_Nf\|_{L^p(\mathbb{R}^n\times I)}\le C_{n,\varphi}[\log(2+N)]^{2/p}\|f\|_{L^p(\mathbb{R}^n)}. \tag*{(2)} ∥ U ( T ) P N f ∥ L p ( R n × I ) ≤ C n , φ [ log ( 2 + N ) ] 2/ p ∥ f ∥ L p ( R n ) . (2) Interpolation and summation over annuli yield Theorem 1.1 (Section 7). For unrestricted data on a full annulus the optimal logarithmic power in (1.2) lies between 1 / p c 1/p_c 1/ p c 2 / p c 2/p_c 2/ p c p c p_c p c 1 / p c 1/p_c 1/ p c
A conserved current and two circles
Consider a finite sum of plane waves with ∣ ξ ∣ ≍ N |\xi|\asymp N ∣ ξ ∣ ≍ N − i ℓ ( 1 , ω ) -i\ell(1,\omega) − i ℓ ( 1 , ω ) ∣ ω ∣ = 1 |\omega|=1 ∣ ω ∣ = 1 ℓ > 0 \ell>0 ℓ > 0 ξ \xi ξ exp [ − ℓ ∣ ξ ∣ ∣ ω − ξ / ∣ ξ ∣ ∣ 2 / 2 ] \exp[-\ell|\xi||\omega-\xi/|\xi||^2/2] exp [ − ℓ ∣ ξ ∣∣ ω − ξ /∣ ξ ∣ ∣ 2 /2 ] ( N ℓ ) − 1 / 2 (N\ell)^{-1/2} ( N ℓ ) − 1/2 N ℓ > 1 N\ell>1 N ℓ > 1
Here is the estimate that the construction must deliver. Set h = N − 1 h=N^{-1} h = N − 1 a = K h a=Kh a = K h K K K p = p c p=p_c p = p c T T T τ ∈ R 2 \tau\in\mathbb{R}^2 τ ∈ R 2 H ( T , s ) H(T,s) H ( T , s ) ∣ τ ∣ = s |\tau|=s ∣ τ ∣ = s F ( T , τ ) \mathcal{F}(T,\tau) F ( T , τ ) h h h ∥ f ∥ 2 = 1 \|f\|_2=1 ∥ f ∥ 2 = 1
∫ 1 2 ∥ U ( T ) f ∥ p p d T ≲ a − 2 ∫ 1 2 H ( T , 2 a ) d T \int_1^2\|U(T)f\|_p^p\,dT\lesssim a^{-2}\int_1^2H(T,2a)\,dT ∫ 1 2 ∥ U ( T ) f ∥ p p d T ≲ a − 2 ∫ 1 2 H ( T , 2 a ) d T ≲ [ log ( 2 / h ) ] 2 a − 2 H ( 0 , 4 a ) ≲ [ log ( 2 / h ) ] 2 ∥ f ∥ p p . \lesssim[\log(2/h)]^2a^{-2}H(0,4a)\lesssim[\log(2/h)]^2\|f\|_p^p. ≲ [ log ( 2/ h ) ] 2 a − 2 H ( 0 , 4 a ) ≲ [ log ( 2/ h ) ] 2 ∥ f ∥ p p . The first and last inequalities compare the energy with the wave. The middle one uses only a scalar differential inequality.
To obtain the energy, fix ( T , τ ) (T,\tau) ( T , τ ) μ > 0 \mu>0 μ > 0 Φ \Phi Φ λ > 0 \lambda>0 λ > 0 H f i b ( λ ∣ Φ ∣ 2 + u ) H_{\mathrm{fib}}(\lambda|\Phi|^2+u) H fib ( λ ∣Φ ∣ 2 + u )
det H f i b ( λ ∣ Φ ∣ 2 + u ) = λ μ , u ∣ ∂ = 0. \det H_{\mathrm{fib} }(\lambda|\Phi|^2+u)=\lambda\mu,\qquad u|_{\partial}=0. det H fib ( λ ∣Φ ∣ 2 + u ) = λ μ , u ∣ ∂ = 0. Here H f i b H_{\mathrm{fib}} H fib n n n n n n Φ \Phi Φ H ( ∣ Φ ∣ 2 ) H(|\Phi|^2) H ( ∣Φ ∣ 2 ) λ ∣ Φ ∣ 2 \lambda|\Phi|^2 λ ∣Φ ∣ 2 λ \lambda λ α I \alpha I α I ( β + α ) α n − 1 (\beta+\alpha)\alpha^{n-1} ( β + α ) α n − 1 β ≍ λ \beta\asymp\lambda β ≍ λ μ ≥ m \mu\ge m μ ≥ m u u u m 1 / ( n − 1 ) m^{1/(n-1)} m 1/ ( n − 1 ) ∫ ( − u ) μ \int(-u)\mu ∫ ( − u ) μ m n / ( n − 1 ) m^{n/(n-1)} m n / ( n − 1 ) μ \mu μ 2 n / ( n − 1 ) 2n/(n-1) 2 n / ( n − 1 )
A correction to ∫ ( − u ) μ \int(-u)\mu ∫ ( − u ) μ ( T , τ ) (T,\tau) ( T , τ ) λ ∣ Φ ∣ 2 \lambda|\Phi|^2 λ ∣Φ ∣ 2 λ ∣ Φ ∣ 2 + u \lambda|\Phi|^2+u λ ∣Φ ∣ 2 + u [5 , 52 ] . The two integrations by parts use conservation of the wave current and of the Hessian cofactor (§ 3 and § 4 ). The cofactor contributes no boundary term: its normal–normal coefficient vanishes. Only the wave flux through the side of the fiber can give a positive contribution to the Laplacian of the energy. A comparison function made from a spatial average of the squared wave bounds F \mathcal{F} F u u u
Δ T , τ F ≤ C s − 2 e − c s / h F , 2 a ≤ s = ∣ τ ∣ ≤ 3 ( x r e f ‘ S e c t i o n 5 ‘ ) . \Delta_{T,\tau}\mathcal{F}\le Cs^{-2}e^{-cs/h}\mathcal{F},\qquad2a\le s=|\tau|\le3\quad({xref}`Section 5`). Δ T , τ F ≤ C s − 2 e − cs / h F , 2 a ≤ s = ∣ τ ∣ ≤ 3 ( x r e f ‘ S ec t i o n 5‘ ) . It remains to compare the circles. The scalar inequality carries a lower bound at the observed times 1 ≤ T ≤ 2 1\le T\le2 1 ≤ T ≤ 2 ∣ τ ∣ = 2 a |\tau|=2a ∣ τ ∣ = 2 a ∣ τ ∣ = 1 |\tau|=1 ∣ τ ∣ = 1 ∣ τ ∣ = 4 a |\tau|=4a ∣ τ ∣ = 4 a p p p § 7 removes the spatial period to prove Theorem 1.2 , then sums the frequencies for the Sobolev estimate.
Related constructions. The bilinear flux of a symmetric hyperbolic system is classical [25 ] . Analytic continuation in an imaginary spacetime direction has also been used to study waves [42 ] . For positive-energy Klein–Gordon waves, Kaiser found a vector conserved in physical spacetime at each fixed imaginary displacement. Integration over displacements gives a current with nonnegative time density [41 Section 4.5, Equations (36)–(40)] . Here the continued wave gives a positive symmetric tensor whose columns have zero divergence in the combined physical and auxiliary variables.
Serre’s compensated integrability estimates pair a positive divergence-free tensor with the Hessian of a real Monge–Ampère potential to bound a power of its determinant [69 Section 5.1] . The usual stress tensor of a wave need not be positive in n ≥ 2 n\ge2 n ≥ 2 [70 Section 4.1] . Our lifted tensor is positive semidefinite, but before averaging it is a sum of four real squares and its determinant vanishes. We use its regularized time component as the prescribed density in a quaternionic determinant equation on each spatial fiber (§ 4 ).
The main proof ends in § 7 ; Appendix A supplies the Dirichlet theorem used in the construction. Sections 9–12 give the applications. Appendix D replaces the determinant equation by a linear Dirichlet problem on a disk in two dimensions. Appendix E extends the current construction to a simple extremal eigenvalue of a real symmetric n n n n n n n n n
A conserved current for a superposition of waves
We use the flux of a symmetric first-order system twice. It selects an imaginary displacement that gives each mode a Gaussian weight; applied to the continued wave, it gives a positive conserved tensor. The modes are summed before taking the square, so their interference remains in the time component of the tensor. We shall apply the construction on a fixed spatial torus: an annulus then contains only finitely many modes, and integration by parts has no boundary in the real spatial variables. § 7 transfers the estimate to R n \mathbb{R}^n R n
The symbol determines the Gaussian
Choose an orthonormal frame X = ( x , b ) ∈ R d × R X=(x,b)\in\mathbb{R}^d\times\mathbb{R} X = ( x , b ) ∈ R d × R d = n − 1 d=n-1 d = n − 1
t = ( T + b ) / 2 , s 0 = ( b − T ) / 2 . t=(T+b)/\sqrt{2},\qquad s_0=(b-T)/\sqrt{2}. t = ( T + b ) / 2 , s 0 = ( b − T ) / 2 . In these null coordinates the wave operator is 2 ∂ t ∂ s 0 + Δ x 2\partial_t\partial_{s_0}+\Delta_x 2 ∂ t ∂ s 0 + Δ x κ > 0 \kappa>0 κ > 0 s 0 s_0 s 0 2 i κ ∂ t + Δ x 2i\kappa\partial_t+\Delta_x 2 iκ ∂ t + Δ x X ⋅ ξ − T ∣ ξ ∣ X\cdot\xi-T|\xi| X ⋅ ξ − T ∣ ξ ∣ ξ b + ∣ ξ ∣ > 0 \xi_b+|\xi|>0 ξ b + ∣ ξ ∣ > 0
u κ , v = c κ , v exp ( i κ ( s 0 + v ⋅ x − 1 2 ∣ v ∣ 2 t ) ) , κ = ξ b + ∣ ξ ∣ 2 , v = ξ x κ . (3) u_{\kappa,v}=c_{\kappa,v}\exp\left(i\kappa\left(s_0+v\cdot x-\frac{1}{2}|v|^2t\right)\right),\qquad\kappa=\frac{\xi_b+|\xi|}{\sqrt{2}},\qquad v=\frac{\xi_x}{\kappa}. \tag*{(3)} u κ , v = c κ , v exp ( iκ ( s 0 + v ⋅ x − 2 1 ∣ v ∣ 2 t ) ) , κ = 2 ξ b + ∣ ξ ∣ , v = κ ξ x . (3) Here i denotes the ordinary Fourier imaginary unit. Throughout the construction u w = ∑ κ , v u κ , v u_{\mathrm{w}}=\sum_{\kappa,v}u_{\kappa,v} u w = ∑ κ , v u κ , v κ > 0 \kappa>0 κ > 0
Write E a b E_{ab} E ab n × n n\times n n × n ( a , b ) (a,b) ( a , b ) d d d X = ( t , x , s 0 ) X=(t,x,s_0) X = ( t , x , s 0 )
C t = 2 E 00 , C x j = E 0 j + E j 0 , C s 0 = − ∑ j = 1 d E j j , P = ∑ μ C μ ∂ X μ . C^t=2E_{00},\qquad C^{x_j}=E_{0j}+E_{j0},\qquad C^{s_0}=-\sum_{j=1}^{d}E_{jj},\qquad\mathcal{P}=\sum_{\mu}C^\mu\partial_{X^\mu}. C t = 2 E 00 , C x j = E 0 j + E j 0 , C s 0 = − j = 1 ∑ d E j j , P = μ ∑ C μ ∂ X μ . Indeed, P ( ∂ s 0 ψ , ∇ x ψ ) t \mathcal{P}(\partial_{s_0}\psi,\nabla_x\psi)^{\mathsf{t}} P ( ∂ s 0 ψ , ∇ x ψ ) t ( 2 ∂ t ∂ s 0 + Δ x ) ψ (2\partial_t\partial_{s_0}+\Delta_x)\psi ( 2 ∂ t ∂ s 0 + Δ x ) ψ p = κ ( − ∣ v ∣ 2 / 2 , v , 1 ) p=\kappa(-|v|^2/2,v,1) p = κ ( − ∣ v ∣ 2 /2 , v , 1 ) C ( p ) = ∑ μ p μ C μ C(p)=\sum_\mu p_\mu C^\mu C ( p ) = ∑ μ p μ C μ
− C ( p ) = κ ( ∣ v ∣ 2 − v t − v I d ) , − w t C ( p ) w = κ ∣ w ′ − v w 0 ∣ 2 . (4) -C(p)=\kappa\begin{pmatrix}|v|^2&-v^{\mathsf{t}}\\-v&I_d\end{pmatrix},\qquad-w^{\mathsf{t}}C(p)w=\kappa|w'-vw_0|^2. \tag*{(4)} − C ( p ) = κ ( ∣ v ∣ 2 − v − v t I d ) , − w t C ( p ) w = κ ∣ w ′ − v w 0 ∣ 2 . (4) In particular, the kernel is spanned by E ( v ) = ( 1 , v ) t E(v)=(1,v)^{\mathsf{t}} E ( v ) = ( 1 , v ) t f s y s = ∑ κ , v κ d / 2 E ( v ) u κ , v f_{\mathrm{sys}}=\sum_{\kappa,v}\kappa^{d/2}E(v)u_{\kappa,v} f sys = ∑ κ , v κ d /2 E ( v ) u κ , v P f s y s = 0 \mathcal{P}f_{\mathrm{sys}}=0 P f sys = 0
To see the Gaussian in physical coordinates, translate ( T , X ) (T,X) ( T , X ) − i ρ ( 1 , ω ) -i\rho(1,\omega) − i ρ ( 1 , ω ) ρ > 0 \rho>0 ρ > 0 ∣ ω ∣ = 1 |\omega|=1 ∣ ω ∣ = 1 ξ \xi ξ
e ρ ( ξ ⋅ ω − ∣ ξ ∣ ) = e − ρ ∣ ξ ∣ ∣ ξ ^ − ω ∣ 2 / 2 , ξ ^ = ξ / ∣ ξ ∣ . e^{\rho(\xi\cdot\omega-|\xi|)}=e^{-\rho|\xi||\widehat{\xi}-\omega|^2/2},\qquad\widehat{\xi}=\xi/|\xi|. e ρ ( ξ ⋅ ω − ∣ ξ ∣ ) = e − ρ ∣ ξ ∣∣ ξ − ω ∣ 2 /2 , ξ = ξ /∣ ξ ∣. For ∣ ξ ∣ ≃ N |\xi| \simeq N ∣ ξ ∣ ≃ N ρ N ≫ 1 \rho N \gg1 ρN ≫ 1 ( ρ N ) − 1 / 2 (\rho N)^{-1/2} ( ρN ) − 1/2
For a real state w = ( w 0 , w ′ ) w=(w_0,w') w = ( w 0 , w ′ ) q ( w ) μ = w t C μ w / 2 q(w)^\mu=w^{\mathsf{t}}C^\mu w/2 q ( w ) μ = w t C μ w /2 ( w 0 2 , w 0 w ′ , − ∣ w ′ ∣ 2 / 2 ) (w_0^2,w_0w',-|w'|^2/2) ( w 0 2 , w 0 w ′ , − ∣ w ′ ∣ 2 /2 ) q ( w ) T = ∣ q ( w ) X ∣ = ( w 0 2 + ∣ w ′ ∣ 2 / 2 ) / 2 q(w)^T=|q(w)^X|=(w_0^2+|w'|^2/2)/\sqrt{2} q ( w ) T = ∣ q ( w ) X ∣ = ( w 0 2 + ∣ w ′ ∣ 2 /2 ) / 2 E ( v ) E(v) E ( v ) q ( E ( v ) ) = ( ∣ ξ ∣ , ξ ) / κ q(E(v))=(|\xi|,\xi)/\kappa q ( E ( v )) = ( ∣ ξ ∣ , ξ ) / κ
q μ = q ( w 1 ) μ + q ( w 2 ) μ = 1 2 ∑ ℓ = 1 2 w ℓ t C μ w ℓ , w 1 , w 2 ∈ R n . (5) q^\mu=q(w_1)^\mu+q(w_2)^\mu=\frac{1}{2}\sum_{\ell=1}^{2}w_\ell^{\mathsf{t}}C^\mu w_\ell,\qquad w_1,w_2\in\mathbb{R}^n. \tag*{(5)} q μ = q ( w 1 ) μ + q ( w 2 ) μ = 2 1 ℓ = 1 ∑ 2 w ℓ t C μ w ℓ , w 1 , w 2 ∈ R n . (5) At X − i q X-iq X − i q e p ⋅ q e^{p\cdot q} e p ⋅ q
z = w 1 + k w 2 = ( R , ζ ) ∈ C k × C k d , C k = { a + k b : a , b ∈ R } , k 2 = − 1. z=w_1+kw_2=(R,\zeta)\in\mathbb{C}_k\times\mathbb{C}_k^d,\qquad\mathbb{C}_k=\{a+kb:a,b\in\mathbb{R}\},\qquad k^2=-1. z = w 1 + k w 2 = ( R , ζ ) ∈ C k × C k d , C k = { a + k b : a , b ∈ R } , k 2 = − 1. Thus q μ = z ∗ C μ z / 2 q^\mu=z^*C^\mu z/2 q μ = z ∗ C μ z /2 ∗ * ∗ (4) gives
e p ⋅ q = exp ( − κ 2 ∣ ζ − v R ∣ 2 ) . (6) e^{p\cdot q}=\exp\left(-\frac{\kappa}{2}|\zeta-vR|^2\right). \tag*{(6)} e p ⋅ q = exp ( − 2 κ ∣ ζ − v R ∣ 2 ) . (6) This explains the normalization in f s y s f_{\mathrm{sys}} f sys κ d / 2 e p ⋅ q \kappa^{d/2}e^{p\cdot q} κ d /2 e p ⋅ q π d \pi^d π d ζ \zeta ζ R R R
Two auxiliary flows
Recall the bilinear conservation identity for symmetric systems, as in Friedrichs [25 ] . For constant real symmetric matrices C μ C^\mu C μ a , b a,b a , b j μ ( a , b ) = a t C μ b j^\mu(a,b)=a^{\mathsf{t}}C^\mu b j μ ( a , b ) = a t C μ b ( P a ) t b + a t P b (\mathcal{P}a)^{\mathsf{t}}b+a^{\mathsf{t}}\mathcal{P}b ( P a ) t b + a t P b q μ q^\mu q μ δ w ℓ \delta w_\ell δ w ℓ j μ ( w ℓ , δ w ℓ ) j^\mu(w_\ell,\delta w_\ell) j μ ( w ℓ , δ w ℓ )
Lemma 2.1. Let P = ∑ μ C μ ∂ X μ \mathcal{P}=\sum_\mu C^\mu\partial_{X^\mu} P = ∑ μ C μ ∂ X μ P f s y s = 0 \mathcal{P}f_{\mathrm{sys}}=0 P f sys = 0 (5) , put, wherever defined,
G = f s y s ( X − i q ) , V ℓ a = ∑ μ ( C μ w ℓ ) a ∂ X μ , D ℓ a = ∂ w ℓ a + i V ℓ a , Z ℓ = ∑ a G a D ℓ a . G=f_{\mathrm{sys}}(X-iq),\qquad V_{\ell a}=\sum_\mu(C^\mu w_\ell)_a\partial_{X^\mu},\qquad D_{\ell a}=\partial_{w_{\ell a}}+iV_{\ell a},\qquad Z_\ell=\sum_a G_aD_{\ell a}. G = f sys ( X − i q ) , V ℓ a = μ ∑ ( C μ w ℓ ) a ∂ X μ , D ℓ a = ∂ w ℓ a + i V ℓ a , Z ℓ = a ∑ G a D ℓ a . In the real coordinates ( X , w 1 , w 2 ) (X,w_1,w_2) ( X , w 1 , w 2 )
B = Re ∑ ℓ = 1 2 Z ℓ ⊗ Z ℓ ‾ (7) B=\operatorname{Re}\sum_{\ell=1}^{2}Z_\ell\otimes\overline{Z_\ell} \tag*{(7)} B = Re ℓ = 1 ∑ 2 Z ℓ ⊗ Z ℓ (7) is symmetric and positive semidefinite, and each of its columns has zero divergence.
Proof. The fields D ℓ a D_{\ell a} D ℓ a X − i q X-iq X − i q [6 Section 1] . Each annihilates every component of X − i q X-iq X − i q D ℓ a G ‾ = 0 D_{\ell a}\overline{G}=0 D ℓ a G = 0 ∂ w ℓ a G = i V ℓ a G \partial_{w_{\ell a}}G=iV_{\ell a}G ∂ w ℓ a G = i V ℓ a G D ℓ a D_{\ell a} D ℓ a
div Z ℓ = 2 i ∑ a V ℓ a G a = 2 i w ℓ t P G = 0. \operatorname{div}Z_\ell=2i\sum_a V_{\ell a}G_a=2iw_\ell^{\mathsf{t}}\mathcal{P}G=0. div Z ℓ = 2 i a ∑ V ℓ a G a = 2 i w ℓ t P G = 0. For the ordinary flat connection, the identity D ℓ a G ‾ = 0 D_{\ell a}\overline{G}=0 D ℓ a G = 0 D m b ‾ \overline{D_{mb}} D mb
∇ Z ℓ Z m ‾ = − i δ ℓ m ∑ μ ( G t C μ G ‾ ) ∂ X μ . \nabla_{Z_\ell}\overline{Z_m}=-i\delta_{\ell m}\sum_\mu(G^{\mathsf{t}}C^\mu\overline{G})\partial_{X^\mu}. ∇ Z ℓ Z m = − i δ ℓ m μ ∑ ( G t C μ G ) ∂ X μ . The vector in the sum is real by symmetry of the matrices. The product rule for div ( Z ℓ ⊗ Z ℓ ‾ ) \operatorname{div}(Z_\ell\otimes\overline{Z_\ell}) div ( Z ℓ ⊗ Z ℓ ) Z ℓ = A ℓ + i B ℓ Z_\ell=A_\ell+iB_\ell Z ℓ = A ℓ + i B ℓ (7) is A ℓ ⊗ A ℓ + B ℓ ⊗ B ℓ . A_\ell\otimes A_\ell+B_\ell\otimes B_\ell. A ℓ ⊗ A ℓ + B ℓ ⊗ B ℓ . □ \square □
This cancellation also identifies the physical current in the lifted variables: the imaginary part of the last differentiation identity at m = ℓ m=\ell m = ℓ [ A ℓ , B ℓ ] = ∑ μ ( G t C μ G ‾ ) ∂ X μ [A_\ell,B_\ell]=\sum_\mu(G^{\mathsf{t}}C^\mu\overline{G})\partial_{X^\mu} [ A ℓ , B ℓ ] = ∑ μ ( G t C μ G ) ∂ X μ B B B
Coordinates adapted to the wave
We want one auxiliary pair for each of the n + 1 n+1 n + 1 X μ + ω μ i X^\mu+\omega_\mu i X μ + ω μ i ω μ ∈ C k \omega_\mu\in\mathbb{C}_k ω μ ∈ C k H \mathbb{H} H i , j , k \mathbf{i},\mathbf{j},\mathbf{k} i , j , k i 2 = j 2 = k 2 = − 1 \mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=-1 i 2 = j 2 = k 2 = − 1 i j = k \mathbf{i}\mathbf{j}=\mathbf{k} ij = k k i = j \mathbf{k}\mathbf{i}=\mathbf{j} ki = j D = span R ( 1 , i , j ) D=\operatorname{span}_{\mathbb{R}}(1,\mathbf{i},\mathbf{j}) D = span R ( 1 , i , j ) C i \mathbb{C}_i C i α + k β \alpha+\mathbf{k}\beta α + k β α , β ∈ C i \alpha,\beta\in\mathbb{C}_i α , β ∈ C i ⟨ u , v ⟩ = Re ( u v ‾ ) \langle u,v\rangle=\operatorname{Re}(u\overline{v}) ⟨ u , v ⟩ = Re ( u v ) ∗ * ∗
The representation we seek for B B B A A A
A μ α , ν β R = Re ( e α ‾ A μ ν e β ) , ( e 0 , e 1 , e 2 ) = ( 1 , i , j ) . (8) A^{\mathbb{R}}_{\mu\alpha,\nu\beta}=\operatorname{Re}(\overline{e_\alpha}A_{\mu\nu}e_\beta),\qquad(e_0,e_1,e_2)=(1,\mathbf{i},\mathbf{j}). \tag*{(8)} A μα , ν β R = Re ( e α A μν e β ) , ( e 0 , e 1 , e 2 ) = ( 1 , i , j ) . (8) If A = χ χ ∗ A=\chi\chi^* A = χ χ ∗ e α ‾ χ μ \overline{e_\alpha}\chi_\mu e α χ μ μ \mu μ Q Q Q − i Q -\mathbf{i}Q − i Q − j Q -\mathbf{j}Q − j Q
In source coordinates, write the four squares of (7) as a Gram matrix by setting Q I = Z I 1 + k Z I 2 Q_I=Z^1_I+\mathbf{k}Z^2_I Q I = Z I 1 + k Z I 2 I I I C i \mathbb{C}_i C i k C i \mathbf{k}\mathbb{C}_i k C i B I J = Re ( Q I Q J ‾ ) B^{IJ}=\operatorname{Re}(Q_I\overline{Q_J}) B I J = Re ( Q I Q J ) M μ a = ( C μ z ) a M_{\mu a}=(C^\mu z)_a M μ a = ( C μ z ) a
Q X μ = ∑ a M μ a i G a , ( Q w 1 a , Q w 2 a ) = ( G a , k G a ) . Q_{X^\mu}=\sum_a M_{\mu a}\mathbf{i}G_a,\qquad(Q_{w_{1a}},Q_{w_{2a}})=(G_a,\mathbf{k}G_a). Q X μ = a ∑ M μ a i G a , ( Q w 1 a , Q w 2 a ) = ( G a , k G a ) . We now choose the target auxiliary coordinates to give the desired rows − i Q X μ , − j Q X μ -\mathbf{i}Q_{X^\mu},-\mathbf{j}Q_{X^\mu} − i Q X μ , − j Q X μ ω μ ( z ) \omega_\mu(z) ω μ ( z ) L μ a = ∂ z a ω μ L_{\mu a}=\partial_{z_a}\omega_\mu L μ a = ∂ z a ω μ c = c 1 + k c 2 c=c_1+\mathbf{k}c_2 c = c 1 + k c 2 ( c 1 , − c 2 ) , ( c 2 , c 1 ) (c_1,-c_2),(c_2,c_1) ( c 1 , − c 2 ) , ( c 2 , c 1 ) ( G a , k G a ) (G_a,\mathbf{k}G_a) ( G a , k G a ) ( c G a , k c G a ) (cG_a,\mathbf{k}cG_a) ( c G a , k c G a ) μ \mu μ ∑ a L μ a G a \sum_aL_{\mu a}G_a ∑ a L μ a G a L = M L=M L = M − i Q X μ -\mathbf{i}Q_{X^\mu} − i Q X μ − i c i = c ‾ -\mathbf{i}c\mathbf{i}=\overline{c} − i c i = c c ∈ C k c\in\mathbb{C}_k c ∈ C k k ( − i Q X μ ) = − j Q X μ \mathbf{k}(-\mathbf{i}Q_{X^\mu})=-\mathbf{j}Q_{X^\mu} k ( − i Q X μ ) = − j Q X μ
The holomorphic flux map. The derivative L μ a = ( C μ z ) a L_{\mu a}=(C^\mu z)_a L μ a = ( C μ z ) a z t C μ z / 2 z^tC^\mu z/2 z t C μ z /2 z ∗ C μ z / 2 z^*C^\mu z/2 z ∗ C μ z /2 n n n n + 1 n+1 n + 1 η ∈ C k \eta\in\mathbb{C}_k η ∈ C k c ∈ R n + 1 c\in\mathbb{R}^{n+1} c ∈ R n + 1
Θ : ( z , η ) ⟼ ω , ω μ = 1 2 z t C μ z + c μ η . \Theta:(z,\eta)\longmapsto\omega,\qquad\omega_\mu=\frac{1}{2}z^tC^\mu z+c^\mu\eta. Θ : ( z , η ) ⟼ ω , ω μ = 2 1 z t C μ z + c μ η . The map fixes the real spacetime coordinates. On a local inverse chart its real volume Jacobian is J = ∣ det C [ M , c ] ∣ 2 J=\lvert\det_{\mathbb{C}}[M,c]\rvert^2 J = ∣ det C [ M , c ] ∣ 2 M μ a = ( C μ z ) a M_{\mu a}=(C^\mu z)_a M μ a = ( C μ z ) a
B ~ ∘ Θ = J − 1 D Θ B D Θ t . (9) \widetilde{B}\circ\Theta=J^{-1}D\Theta\,B\,D\Theta^t. \tag*{(9)} B ∘ Θ = J − 1 D Θ B D Θ t . (9) The two differentials transform the Gram rows as above, and J − 1 J^{-1} J − 1
B ~ = ( χ ~ χ ~ ∗ ) R , χ ~ μ ∘ Θ = J − 1 / 2 ∑ a M μ a i G a . (10) \widetilde{B}=(\widetilde{\chi}\widetilde{\chi}^*)^{\mathbb{R}},\qquad\widetilde{\chi}_\mu\circ\Theta=J^{-1/2}\sum_a M_{\mu a}\mathbf{i}G_a. \tag*{(10)} B = ( χ χ ∗ ) R , χ μ ∘ Θ = J − 1/2 a ∑ M μ a i G a . (10) As a function of ω \omega ω ( ∂ ω M ) χ ~ = 0 (\partial_\omega M)\widetilde{\chi}=0 ( ∂ ω M ) χ = 0 z z z η \eta η M M M ( ∂ ω χ ~ ) = 0 (\partial_\omega\widetilde{\chi})=0 ( ∂ ω χ ) = 0
To verify conservation, test the K K K φ \varphi φ (9) , the integral for the weak divergence becomes − ∑ I , L ∫ B I L ( ∂ L Θ K ) ∂ I ( φ ∘ Θ ) -\sum_{I,L}\int B^{IL}(\partial_L\Theta^K)\partial_I(\varphi\circ\Theta) − ∑ I , L ∫ B I L ( ∂ L Θ K ) ∂ I ( φ ∘ Θ )
J ( div B ~ ) K ∘ Θ = ∑ L ( ∂ L Θ K ) ( div B ) L + ∑ I , L B I L ∂ I ∂ L Θ K . J(\operatorname{div}\widetilde{B})^K\circ\Theta=\sum_L(\partial_L\Theta^K)(\operatorname{div}B)^L+\sum_{I,L}B^{IL}\partial_I\partial_L\Theta^K. J ( div B ) K ∘ Θ = L ∑ ( ∂ L Θ K ) ( div B ) L + I , L ∑ B I L ∂ I ∂ L Θ K . The first sum vanishes by Lemma 2.1. In the second sum the two diagonal state blocks of B B B ( Re G a G b ‾ ) a b (\operatorname{Re}G_a\overline{G_b})_{ab} ( Re G a G b ) ab ∂ w 1 a w 1 b Θ K + ∂ w 2 a w 2 b Θ K = 0 \partial_{w_{1a}w_{1b}}\Theta^K+\partial_{w_{2a}w_{2b}}\Theta^K=0 ∂ w 1 a w 1 b Θ K + ∂ w 2 a w 2 b Θ K = 0 Θ \Theta Θ
The wave in these coordinates. For the wave matrices, add η \eta η s 0 s_0 s 0
Θ : ( R , ζ , η ) ⟼ ( Y , P , S 1 ) = ( R 2 , R ζ , η − ζ ⋅ ζ / 2 ) . (11) \Theta:(R,\zeta,\eta)\longmapsto(Y,P,S_1)=(R^2,R\zeta,\eta-\zeta\cdot\zeta/2). \tag*{(11)} Θ : ( R , ζ , η ) ⟼ ( Y , P , S 1 ) = ( R 2 , R ζ , η − ζ ⋅ ζ /2 ) . (11) If ( τ , P , Z ) (\tau,P,Z) ( τ , P , Z ) ( T , x , b ) (T,x,b) ( T , x , b ) Y = ( τ + Z ) / 2 Y=(\tau+Z)/\sqrt{2} Y = ( τ + Z ) / 2 S 1 = ( Z − τ ) / 2 S_1=(Z-\tau)/\sqrt{2} S 1 = ( Z − τ ) / 2 (11) on Y ≠ 0 Y\ne0 Y = 0 R = Y R=\sqrt{Y} R = Y ζ = P / R \zeta=P/R ζ = P / R R R R
W = P Y = ζ R , Φ = S 1 + P ⋅ P 2 Y = η , y = ∣ Y ∣ . (12) W=\frac{P}{Y}=\frac{\zeta}{R},\qquad\Phi=S_1+\frac{P\cdot P}{2Y}=\eta,\qquad y=|Y|. \tag*{(12)} W = Y P = R ζ , Φ = S 1 + 2 Y P ⋅ P = η , y = ∣ Y ∣. (12) All dots between complex vectors here are bilinear; ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣ Θ \Theta Θ 2 R d + 1 2R^{d+1} 2 R d + 1
J = 4 ∣ R ∣ 2 d + 2 . (13) J=4|R|^{2d+2}. \tag*{(13)} J = 4∣ R ∣ 2 d + 2 . (13) Under the derivative M M M E ( v ) E(v) E ( v )
M E ( v ) = ( 2 R , ζ + R v , − ζ ⋅ v ) t = 2 R V v , V v = ( 1 , ( v + W ) / 2 , − W ⋅ v / 2 ) t . (14) ME(v)=(2R,\zeta+Rv,-\zeta\cdot v)^{\mathsf{t}}=2RV^v,\qquad V^v=(1,(v+W)/2,-W\cdot v/2)^{\mathsf{t}}. \tag*{(14)} M E ( v ) = ( 2 R , ζ + R v , − ζ ⋅ v ) t = 2 R V v , V v = ( 1 , ( v + W ) /2 , − W ⋅ v /2 ) t . (14) In particular Φ \Phi Φ ℓ ( w ) = ( − w ⋅ w / 2 , w , 1 ) \ell(w)=(-w\cdot w/2,w,1) ℓ ( w ) = ( − w ⋅ w /2 , w , 1 ) κ ℓ ( v ) \kappa\ell(v) κ ℓ ( v ) ℓ ( w ) V v = − ( w − W ) ⋅ ( w − v ) / 2 \ell(w)V^v=-(w-W)\cdot(w-v)/2 ℓ ( w ) V v = − ( w − W ) ⋅ ( w − v ) /2 ℓ ( v ) \ell(v) ℓ ( v ) ∂ Φ = ℓ ( W ) \partial\Phi=\ell(W) ∂ Φ = ℓ ( W ) W = P / Y W=P/Y W = P / Y
( ∂ Φ ) V v = 0 , ( ∂ W ) V v = v − W 2 Y . (15) (\partial\Phi)V^v=0,\qquad(\partial W)V^v=\frac{v-W}{2Y}. \tag*{(15)} ( ∂ Φ ) V v = 0 , ( ∂ W ) V v = 2 Y v − W . (15) These are complex differentials in ( Y , P , S 1 ) (Y,P,S_1) ( Y , P , S 1 ) Φ \Phi Φ W W W v − W v-W v − W
For e = Y / y e=Y/y e = Y / y ι e ( α + i β ) = α + e i β \iota_e(\alpha+i\beta)=\alpha+e i\beta ι e ( α + i β ) = α + e i β H \mathbb{H} H i c = c ‾ i ic=\overline{c}i i c = c i c ∈ C k c\in\mathbb{C}_k c ∈ C k ( e i ) 2 = − 1 (ei)^2=-1 ( e i ) 2 = − 1
m κ , v = ( κ / y ) d / 2 exp ( − κ y 2 ∣ W − v ∣ 2 ) . m_{\kappa,v}=(\kappa/y)^{d/2}\exp\left(-\frac{\kappa y}{2}|W-v|^2\right). m κ , v = ( κ / y ) d /2 exp ( − 2 κ y ∣ W − v ∣ 2 ) . The factor y − d / 2 y^{-d/2} y − d /2
Proposition 2.2 (The wave current). For any finite superposition (3) , set
χ = ∑ κ , v V v ι e ( u κ , v ‾ ) m κ , v , A 0 = χ χ ∗ . (16) \chi=\sum_{\kappa,v}V^v\iota_e(\overline{u_{\kappa,v}})m_{\kappa,v},\qquad A^0=\chi\chi^*. \tag*{(16)} χ = κ , v ∑ V v ι e ( u κ , v ) m κ , v , A 0 = χ χ ∗ . (16) On Y ≠ 0 Y\ne0 Y = 0 A 0 A^0 A 0 ( ∂ Φ ) A 0 = 0 (\partial\Phi)A^0=0 ( ∂ Φ ) A 0 = 0 ( A 0 ) R (A^0)_{\mathbb{R}} ( A 0 ) R
V T v = ( 1 + W ⋅ v / 2 ) / 2 , A T T 0 = ∣ χ T ∣ 2 . V_T^v=(1+W\cdot v/2)/\sqrt{2},\qquad A^0_{TT}=|\chi_T|^2. V T v = ( 1 + W ⋅ v /2 ) / 2 , A T T 0 = ∣ χ T ∣ 2 . Proof. We identify the column in (10) on one inverse chart. By (6) and (14) , the contribution of a single mode to χ ~ \widetilde{\chi} χ
V v g i u κ , v m κ , v , g = R / ∣ R ∣ . V^{v}g i u_{\kappa,v}m_{\kappa,v}, \qquad g=R/|R|. V v g i u κ , v m κ , v , g = R /∣ R ∣. Indeed ∣ ζ − v R ∣ 2 = y ∣ W − v ∣ 2 |\zeta-vR|^{2}=y|W-v|^{2} ∣ ζ − v R ∣ 2 = y ∣ W − v ∣ 2 2 ∣ R ∣ / J = y − d / 2 2|R|/\sqrt{J}=y^{-d/2} 2∣ R ∣/ J = y − d /2 ( g i ) − 1 (gi)^{-1} ( g i ) − 1 g g g i i i g 2 i = e i g^{2}i=ei g 2 i = e i i i i (16) , with V v V^{v} V v Y Y Y (15) . The change back to physical coordinates acts by the same real orthogonal matrix on each of the three coordinate slots, so it preserves the column divergence and the row–column pairing. Finally T = ( t − s 0 ) / 2 T=(t-s_{0})/\sqrt{2} T = ( t − s 0 ) / 2 V T v V^{v}_{T} V T v
For a single κ \kappa κ W W W m κ , v m_{\kappa,v} m κ , v κ v \kappa v κ v κ W \kappa W κW V t v = 1 V^{v}_{t}=1 V t v = 1 A t t 0 A^{0}_{tt} A tt 0 [39 ] ; see [45 Section 2.3] . For the full wave, (16) sums the amplitudes over κ \kappa κ
The time component on a frequency cap
At a fixed auxiliary point write V T v = b 0 + k b 1 V^{v}_{T}=b_{0}+\mathbf{k}b_{1} V T v = b 0 + k b 1
b 0 = 1 + Re ( W ) ⋅ v / 2 2 , b 1 = Im k ( W ) ⋅ v 2 2 . b_{0}=\frac{1+\operatorname{Re}(W)\cdot v/2}{\sqrt{2}},\qquad b_{1}=\frac{\operatorname{Im}_{\mathbf{k}}(W)\cdot v}{2\sqrt{2}}. b 0 = 2 1 + Re ( W ) ⋅ v /2 , b 1 = 2 2 Im k ( W ) ⋅ v . Here Im k ( α + k β ) = β \operatorname{Im}_{\mathbf{k}}(\alpha+\mathbf{k}\beta)=\beta Im k ( α + k β ) = β S ℓ S_{\ell} S ℓ (3) to b ℓ m κ , v u κ , v b_{\ell}m_{\kappa,v}u_{\kappa,v} b ℓ m κ , v u κ , v (16) ,
χ ~ T = ι e ( S 0 u w ) + k ι e ( S 1 u w ) , A T T 0 = ∣ S 0 u w ∣ 2 + ∣ S 1 u w ∣ 2 . (17) \widetilde{\chi}_{T}=\iota_{e}(S_{0}u_{w})+\mathbf{k}\,\iota_{e}(S_{1}u_{w}),\qquad A^{0}_{TT}=|S_{0}u_{w}|^{2}+|S_{1}u_{w}|^{2}. \tag*{(17)} χ T = ι e ( S 0 u w ) + k ι e ( S 1 u w ) , A T T 0 = ∣ S 0 u w ∣ 2 + ∣ S 1 u w ∣ 2 . (17) Indeed the two terms lie in the conjugates of the orthogonal planes C i \mathbb{C}_{i} C i k C i \mathbf{k}\mathbb{C}_{i} k C i g g g e e e C k \mathbb{C}_{\mathbf{k}} C k W = 0 W=0 W = 0 2 A T T 0 2A^{0}_{TT} 2 A T T 0 ( κ / y ) d / 2 exp ( − κ y ∣ v ∣ 2 / 2 ) (\kappa/y)^{d/2}\exp(-\kappa y|v|^{2}/2) ( κ / y ) d /2 exp ( − κ y ∣ v ∣ 2 /2 )
For the estimates take u w = e − i T ∣ D ∣ f u_{w}=e^{-iT|D|}f u w = e − i T ∣ D ∣ f R n / ( 100 Z n ) \mathbb{R}^{n}/(100\mathbb{Z}^{n}) R n / ( 100 Z n ) b b b 0 < h ≤ 1 0<h\leq1 0 < h ≤ 1 f ^ \widehat{f} f c 0 / h ≤ ∣ ξ ∣ ≤ C 0 / h c_{0}/h\leq|\xi|\leq C_{0}/h c 0 / h ≤ ∣ ξ ∣ ≤ C 0 / h ∣ v ∣ ≤ γ |v|\leq\gamma ∣ v ∣ ≤ γ 0 < c 0 < C 0 0<c_{0}<C_{0} 0 < c 0 < C 0 γ > 0 \gamma>0 γ > 0 h h h ∣ ξ ∣ = κ ( 1 + ∣ v ∣ 2 / 2 ) / 2 |\xi|=\kappa(1+|v|^{2}/2)/\sqrt{2} ∣ ξ ∣ = κ ( 1 + ∣ v ∣ 2 /2 ) / 2 κ ≍ h − 1 \kappa\asymp h^{-1} κ ≍ h − 1 y ≍ h y\asymp h y ≍ h ∣ W ∣ , ∣ v ∣ ≤ 1 / 10 |W|,|v|\leq1/10 ∣ W ∣ , ∣ v ∣ ≤ 1/10 b 0 b_{0} b 0 h d S 0 h^{d}S_{0} h d S 0 y / h y/h y / h ∣ W − v ∣ |W-v| ∣ W − v ∣ O ( h − n ) O(h^{-n}) O ( h − n ) ∑ κ , v ∣ c κ , v ∣ ≲ h − n / 2 ∥ f ∥ 2 \sum_{\kappa,v}|c_{\kappa,v}|\lesssim h^{-n/2}\|f\|_{2} ∑ κ , v ∣ c κ , v ∣ ≲ h − n /2 ∥ f ∥ 2 ∥ f ∥ 2 = 1 \|f\|_{2}=1 ∥ f ∥ 2 = 1
The determinant and integration by parts
The wave current in Proposition 2.2 is positive and has divergence-free real columns. We shall pair it with a Hessian. The essential pairing is the ordinary real contraction A : H ( v ) = A R : D 2 v A:H(v)=A^{\mathbb{R}}:D^{2}v A : H ( v ) = A R : D 2 v n n n β \beta β ϵ I \epsilon I ϵ I ( β + ϵ ) ϵ n − 1 (\beta+\epsilon)\epsilon^{n-1} ( β + ϵ ) ϵ n − 1 p c = 2 n / ( n − 1 ) p_c = 2n/(n-1) p c = 2 n / ( n − 1 )
The Hessian in real and complex coordinates
Use the triples z a = x a + P a i ∈ D z_a = x_a + P_ai \in\mathcal{D} z a = x a + P a i ∈ D § 2 , with P a ∈ C k P_a \in\mathbb{C}_k P a ∈ C k m m m z a = ∑ α = 0 k − 1 e α z a α z_a = \sum_{\alpha=0}^{k-1} e_\alpha z_{a\alpha} z a = ∑ α = 0 k − 1 e α z a α k = 3 k = 3 k = 3 4 4 4 ( e 0 , e 1 , e 2 , e 3 ) = ( 1 , i , j , k ) (e_0,e_1,e_2,e_3) = (1,i,\mathbf{j},\mathbf{k}) ( e 0 , e 1 , e 2 , e 3 ) = ( 1 , i , j , k ) z a α z_{a\alpha} z a α C 2 C^2 C 2 u u u ∂ a α = ∂ / ∂ z a α \partial_{a\alpha} = \partial/\partial z_{a\alpha} ∂ a α = ∂ / ∂ z a α
G a ( u ) = ∑ α e α ∂ a α u , H a b ( u ) = ∑ α , β e α e β ‾ ∂ a α ∂ b β u . (18) G_a(u) = \sum_\alpha e_\alpha\partial_{a\alpha}u,\qquad H_{ab}(u) = \sum_{\alpha,\beta} e_\alpha\overline{e_\beta}\,\partial_{a\alpha}\partial_{b\beta}u. \tag*{(18)} G a ( u ) = α ∑ e α ∂ a α u , H ab ( u ) = α , β ∑ e α e β ∂ a α ∂ b β u . (18) Thus G ( u ) G(u) G ( u ) ∇ \nabla ∇ Δ \Delta Δ ∂ n \partial_n ∂ n
Lemma 3.1 (Hessian calculus). In either number of slots, H ( u ) H(u) H ( u ) tr H ( u ) = Δ u \operatorname{tr} H(u) = \Delta u tr H ( u ) = Δ u H ( u ) H(u) H ( u ) ξ ξ t \xi\xi^t ξ ξ t g g ∗ gg^* g g ∗ g a = ∑ α e α ξ a α g_a = \sum_\alpha e_\alpha\xi_{a\alpha} g a = ∑ α e α ξ a α
H ( u v ) = u H ( v ) + v H ( u ) + G ( u ) G ( v ) ∗ + G ( v ) G ( u ) ∗ . (19) H(uv) = uH(v) + vH(u) + G(u)G(v)^* + G(v)G(u)^*. \tag*{(19)} H ( uv ) = u H ( v ) + v H ( u ) + G ( u ) G ( v ) ∗ + G ( v ) G ( u ) ∗ . (19) In four slots H H H
If u u u P ∈ C k m P \in\mathbb{C}_k^m P ∈ C k m
G a ( u ) = 2 ( ∂ P ‾ a u ) i , H a b ( u ) = 4 ∂ P ‾ a ∂ P b u . (20) G_a(u) = 2(\partial_{\overline{P}_a}u)i,\qquad H_{ab}(u) = 4\partial_{\overline{P}_a}\partial_{P_b}u. \tag*{(20)} G a ( u ) = 2 ( ∂ P a u ) i , H ab ( u ) = 4 ∂ P a ∂ P b u . (20) In particular, a holomorphic map F = ( F 1 , … , F l ) F = (F_1,\ldots,F_l) F = ( F 1 , … , F l ) J ν a = ∂ P a F ν J_{\nu a} = \partial_{P_a}F_\nu J ν a = ∂ P a F ν
H ( ∑ ν ∣ F ν ∣ 2 ) = 4 J ∗ J . (21) H\left(\sum_\nu|F_\nu|^2\right) = 4J^*J. \tag*{(21)} H ( ν ∑ ∣ F ν ∣ 2 ) = 4 J ∗ J . (21) For every smooth real function ρ \rho ρ F F F G ( ρ ∘ F ) G(\rho\circ F) G ( ρ ∘ F ) J ∗ J^* J ∗
Proof. Interchanging the real derivatives gives Hermitian symmetry. In a diagonal entry the terms with α ≠ β \alpha\ne\beta α = β ξ ξ t \xi\xi^t ξ ξ t g g ∗ gg^* g g ∗ U U U U ∗ U^* U ∗ H ( u ∘ U ) = U ∗ ( H ( u ) ∘ U ) U H(u\circ U) = U^*(H(u)\circ U)U H ( u ∘ U ) = U ∗ ( H ( u ) ∘ U ) U
For P a = p a + k q a P_a = p_a + kq_a P a = p a + k q a 2 ∂ P ‾ a = ∂ p a + k ∂ q a 2\partial_{\overline{P}_a} = \partial_{p_a} + k\partial_{q_a} 2 ∂ P a = ∂ p a + k ∂ q a k i = j ki = j k i = j H a b H_{ab} H ab 1 , − k , k , 1 1,-k,k,1 1 , − k , k , 1 ( p a p b , p a q b , q a p b , q a q b ) (p_ap_b,p_aq_b,q_ap_b,q_aq_b) ( p a p b , p a q b , q a p b , q a q b ) G ( ρ ∘ F ) = 2 J ∗ ( ∂ F ‾ ν ρ ) ν i G(\rho\circ F) = 2J^*(\partial_{\overline{F}_\nu}\rho)\nu i G ( ρ ∘ F ) = 2 J ∗ ( ∂ F ν ρ ) ν i ρ \rho ρ F F F
The Moore cofactor
For Hermitian quaternionic matrices use the real inner product A : B = Re tr ( A B ) A:B = \operatorname{Re}\operatorname{tr}(AB) A : B = Re tr ( A B ) A : ( v v ∗ ) = v ∗ A v A:(vv^*) = v^*Av A : ( v v ∗ ) = v ∗ A v det A \det A det A m m m [2 Section 1.1, especially Theorems 1.1.8–1.1.9] . The next identities apply also to the full Hessian in the energy argument, whose spatial block is positive definite but whose time entry need not have a sign.
Lemma 3.2 (Matrix identities). The Moore determinant is a homogeneous polynomial of degree m m m C ( A ) C(A) C ( A )
C ( A ) : B = d d t det ( A + t B ) ∣ t = 0 . (22) C(A) : B = \left.\frac{d}{dt}\det(A+tB)\right|_{t=0}. \tag*{(22)} C ( A ) : B = d t d det ( A + tB ) t = 0 . (22) It is polynomial of degree m − 1 m-1 m − 1
C ( A ) = ( det A ) A − 1 ( A > 0 ) . (23) C(A)=(\det A)A^{-1}\qquad(A>0). \tag*{(23)} C ( A ) = ( det A ) A − 1 ( A > 0 ) . (23) If A ≥ 0 A\geq0 A ≥ 0 C ( A ) ≥ 0 C(A)\geq0 C ( A ) ≥ 0 A A A C ( A ) C(A) C ( A ) A A A log det \log\det log det
For a unit vector v ∈ H m v\in\mathbb{H}^{m} v ∈ H m A ∣ v ⊥ A|_{v^\perp} A ∣ v ⊥ A A A t t t
v ∗ C ( A ) v = det ( A ∣ v ⊥ ) , det ( A + t v v ∗ ) = det A + t v ∗ C ( A ) v . (24) v^*C(A)v=\det(A|_{v^\perp}),\qquad\det(A+tvv^*)=\det A+t\,v^*C(A)v. \tag*{(24)} v ∗ C ( A ) v = det ( A ∣ v ⊥ ) , det ( A + t v v ∗ ) = det A + t v ∗ C ( A ) v . (24) The determinant on the zero-dimensional space is one. Finally, for a Hermitian M = ( ℓ c ∗ c D ) M=\begin{pmatrix}\ell&c^*\\c&D\end{pmatrix} M = ( ℓ c c ∗ D ) D > 0 D>0 D > 0 σ = ℓ − c ∗ D − 1 c \sigma=\ell-c^*D^{-1}c σ = ℓ − c ∗ D − 1 c v 0 v_0 v 0
det M = ( det D ) σ , M − σ v 0 v 0 ∗ ≥ 0. (25) \det M=(\det D)\sigma,\qquad M-\sigma v_0v_0^*\geq0. \tag*{(25)} det M = ( det D ) σ , M − σ v 0 v 0 ∗ ≥ 0. (25) Proof. A Hermitian matrix has a quaternionic orthonormal eigenbasis: maximize v ∗ A v v^*Av v ∗ A v Re tr ( A j ) \operatorname{Re}\operatorname{tr}(A^j) Re tr ( A j ) 1 ≤ j ≤ m 1\leq j\leq m 1 ≤ j ≤ m (22) with the stated degrees. In the real representation on R 4 m \mathbb{R}^{4m} R 4 m A > 0 A>0 A > 0 log det A \log\det A log det A (23) , monotonicity and concavity. On a diagonal matrix each cofactor eigenvalue is the product of the other m − 1 m-1 m − 1 (23) , then for all diagonal matrices by polynomiality. Unitary invariance proves the claims for singular A A A
Completing the square shows that the form of M − σ v 0 v 0 ∗ M-\sigma v_0v_0^* M − σ v 0 v 0 ∗ ( b , w ) (b,w) ( b , w ) ( w + D − 1 c b ) ∗ D ( w + D − 1 c b ) (w+D^{-1}cb)^*D(w+D^{-1}cb) ( w + D − 1 c b ) ∗ D ( w + D − 1 c b ) σ > 0 \sigma>0 σ > 0 ( det M ) 4 = ( det D ) 4 σ 4 (\det M)^4=(\det D)^4\sigma^4 ( det M ) 4 = ( det D ) 4 σ 4 det M = ( det D ) σ \det M=(\det D)\sigma det M = ( det D ) σ ℓ \ell ℓ σ \sigma σ v v v t v v ∗ tvv^* t v v ∗ ℓ \ell ℓ A A A t det ( A ∣ v ⊥ ) t\det(A|_{v^\perp}) t det ( A ∣ v ⊥ ) t t t (24) ; polynomiality extends both to every Hermitian A A A m = 1 m=1 m = 1
Column divergence and the boundary term
In k k k (8) of a Hermitian field A ( z ) A(z) A ( z )
A α α , β β R = Re ( e ˉ α A α β e β ) . (26) A^{\mathbb{R}}_{\alpha\alpha,\beta\beta}=\operatorname{Re}(\bar{e}_{\alpha}A_{\alpha\beta}e_{\beta}). \tag*{(26)} A α α , β β R = Re ( e ˉ α A α β e β ) . (26) This symmetric real matrix represents the Hermitian form on the encoded real vectors: ( ∇ v ) ∗ A R ∇ v = Re ( G ( v ) ∗ A G ( u ) ) (\nabla v)^{*}A^{\mathbb{R}}\nabla v=\operatorname{Re}(G(v)^*AG(u)) ( ∇ v ) ∗ A R ∇ v = Re ( G ( v ) ∗ A G ( u )) A : H ( u ) = A R : D 2 u A:H(u)=A^{\mathbb{R}}:D^2u A : H ( u ) = A R : D 2 u
A : H ( u ) = div ( A R ∇ u ) . (27) A:H(u)=\operatorname{div}(A^{\mathbb{R}}\nabla u). \tag*{(27)} A : H ( u ) = div ( A R ∇ u ) . (27) One sufficient condition is the quaternionic row identity
∑ α , a e ˉ α ∂ a α A a b = 0 ( 1 ≤ b ≤ m ) : (28) \sum_{\alpha,a} \bar{e}_{\alpha}\partial_{a\alpha}A_{ab}=0 \quad(1\le b\le m): \tag*{(28)} α , a ∑ e ˉ α ∂ a α A ab = 0 ( 1 ≤ b ≤ m ) : (28) multiply on the right by e β e_{\beta} e β A G ( w ) = 0 AG(w)=0 A G ( w ) = 0 w w w w A wA w A A A A G ( w ) ∗ A = 0 G(w)^*A=0 G ( w ) ∗ A = 0
Alesker showed that the real tensor representing the Moore cofactor of a positive definite quaternionic Hessian has divergence-free columns. This puts the linearized Monge–Ampère operator in divergence form [4 Lemma 4.6] . The following polynomial proof also covers the three slots used by the wave current and allows an indefinite or singular Hessian.
Lemma 3.3 (Cofactor divergence). For k = 3 k=3 k = 3 4 4 4 u u u
∑ α , a e ˉ α ∂ a α C ( H ( u ) ) a b = 0 ( 1 ≤ b ≤ m ) . (29) \sum_{\alpha,a}\bar{e}_{\alpha}\partial_{a\alpha}C(H(u))_{ab}=0 \quad(1\le b\le m). \tag*{(29)} α , a ∑ e ˉ α ∂ a α C ( H ( u ) ) ab = 0 ( 1 ≤ b ≤ m ) . (29) Proof. For m = 1 m=1 m = 1 j 0 = m − 1 j_0=m-1 j 0 = m − 1 M 1 , … , M j 0 M_1,\ldots,M_{j_0} M 1 , … , M j 0 t 1 ⋯ t j 0 t_1\cdots t_{j_0} t 1 ⋯ t j 0 C ( ∑ j t j M j ) C(\sum_j t_jM_j) C ( ∑ j t j M j ) Q ( M 1 , … , M j 0 ) Q(M_1,\ldots,M_{j_0}) Q ( M 1 , … , M j 0 ) j 0 ! C ( M 1 ) j_0!C(M_1) j 0 ! C ( M 1 )
Take M j = g j g j ∗ M_j=g_jg_j^* M j = g j g j ∗ t j > 0 t_j>0 t j > 0 ∑ j t j g j g j ∗ \sum_jt_jg_jg_j^* ∑ j t j g j g j ∗ j 0 j_0 j 0 g j g_j g j t t t Q = Q ( g 1 g 1 ∗ , … , g j 0 g j 0 ∗ ) Q=Q(g_1g_1^*,\ldots,g_{j_0}g_{j_0}^*) Q = Q ( g 1 g 1 ∗ , … , g j 0 g j 0 ∗ ) g j g_j g j g j ∗ Q = 0 g_j^*Q=0 g j ∗ Q = 0 Q Q Q
Test the polarized identity on u j ( z ) = exp ( ξ j ⋅ z ) u_j(z)=\exp(\xi_j\cdot z) u j ( z ) = exp ( ξ j ⋅ z ) ξ j \xi_j ξ j ( g j ) a = ∑ α e α ( ξ j ) a α (g_j)_a=\sum_\alpha e_\alpha(\xi_j)_{a\alpha} ( g j ) a = ∑ α e α ( ξ j ) a α Q ( H ( u 1 ) , … , H ( u j 0 ) ) = exp ( ( ∑ j ξ j ) ⋅ z ) Q Q(H(u_1),\ldots,H(u_{j_0}))=\exp((\sum_j\xi_j)\cdot z)Q Q ( H ( u 1 ) , … , H ( u j 0 )) = exp (( ∑ j ξ j ) ⋅ z ) Q ( ∑ j g j ) ∗ Q = 0 (\sum_jg_j)^*Q=0 ( ∑ j g j ) ∗ Q = 0 u u u (29) .
Here is the boundary convention used for both the current and the cofactor. On a smooth domain Ω = { r < 0 } \Omega=\{r<0\} Ω = { r < 0 } ∇ r ≠ 0 \nabla r\ne0 ∇ r = 0 ∂ Ω \partial\Omega ∂ Ω n = ∇ r / ∣ ∇ r ∣ n=\nabla r/|\nabla r| n = ∇ r /∣∇ r ∣ ν = G ( r ) / ∣ ∇ r ∣ \nu=G(r)/|\nabla r| ν = G ( r ) /∣∇ r ∣ (27) , Green’s identity for a field with divergence-free real columns is
∫ Ω ( v A : H ( u ) − u A : H ( v ) ) = ∫ ∂ Ω Re ( v ν ∗ A G ( u ) − u ν ∗ A G ( v ) ) d S . (30) \int_{\Omega}(vA:H(u)-uA:H(v))=\int_{\partial\Omega}\operatorname{Re}(v\nu^*AG(u)-u\nu^*AG(v))\,dS. \tag*{(30)} ∫ Ω ( v A : H ( u ) − u A : H ( v )) = ∫ ∂ Ω Re ( v ν ∗ A G ( u ) − u ν ∗ A G ( v )) d S . (30) Volume and surface measure are Euclidean. The same formula holds in periodic coordinates, with test functions of compact support in any unbounded directions.
If u u u ∂ Ω \partial\Omega ∂ Ω G ( u ) = ν ∂ n u G(u)=\nu\,\partial_nu G ( u ) = ν ∂ n u u = ρ r u=\rho r u = ρ r ρ \rho ρ r r r u u u (19) then vanish on compression, giving
H ( u ) ∣ ν ⊥ = ρ H ( r ) ∣ ν ⊥ on ∂ Ω . (31) H(u)|_{\nu^\perp}=\rho H(r)|_{\nu^\perp}\quad\text{on }\partial\Omega. \tag*{(31)} H ( u ) ∣ ν ⊥ = ρ H ( r ) ∣ ν ⊥ on ∂ Ω. (31) In four slots this calculation gives the invariance of the quaternionic Levi form under a change of defining function [3 ] . Functions independent of the fourth slot give the same formula in three slots. In particular the normal flux coefficient ν ∗ C ( H ( u ) ) ν \nu^*C(H(u))\nu ν ∗ C ( H ( u )) ν (24) . This is why the determinant boundary term can be evaluated without knowing the second derivative of u u u
The determinant energy and its boundary flux
The time component of the wave current is a sum of squares. After regularization it gives the density for a spatial Dirichlet potential: a lower bound on the density throughout a ball forces a definite depth at its center. To compare that depth at different times, we need a scalar energy on the three real coordinates z = ( T , τ ) ∈ R × C k z=(T,\tau)\in\mathbb{R}\times\mathbb{C}_k z = ( T , τ ) ∈ R × C k
Keep the annular wave on the torus from § 2 , normalized by ∥ f ∥ 2 = 1 \|f\|_2=1 ∥ f ∥ 2 = 1 w 0 = 1 / 10 w_0=1/10 w 0 = 1/10 ∣ v ∣ ≤ γ |v|\leq\gamma ∣ v ∣ ≤ γ γ < w 0 / 4 \gamma<w_0/4 γ < w 0 /4 s = ∣ τ ∣ s=|\tau| s = ∣ τ ∣ a = K h a=Kh a = K h K ≥ 1 K\geq1 K ≥ 1 h h h 4 a ≤ 1 4a\leq1 4 a ≤ 1 a < s < 4 a<s<4 a < s < 4
The geometry of the fibers
At fixed z z z X X X ( P , Z ) ∈ C k d × C k (P,Z)\in\mathbb{C}_k^d\times\mathbb{C}_k ( P , Z ) ∈ C k d × C k
Ω z : r = ∣ W ∣ 2 w 0 2 + 100 ∣ Q ∣ 2 − 1 < 0 , Q = Φ / a . (32) \Omega_z:\qquad r=\frac{|W|^2}{w_0^2}+100|Q|^2-1<0,\qquad Q=\Phi/a. \tag*{(32)} Ω z : r = w 0 2 ∣ W ∣ 2 + 100∣ Q ∣ 2 − 1 < 0 , Q = Φ/ a . (32) The shape of the fiber reflects two properties of the current from Proposition 2.2: it is tangent to the level sets of Φ \Phi Φ W = v W=v W = v v v v W W W ( W , Q ) (W,Q) ( W , Q ) Y = ( τ + Z ) / 2 Y=(\tau+Z)/\sqrt{2} Y = ( τ + Z ) / 2 W = P / Y W=P/Y W = P / Y Φ = ( Z − τ ) / 2 + P ⋅ P / ( 2 Y ) \Phi=(Z-\tau)/\sqrt{2}+P\cdot P/(2Y) Φ = ( Z − τ ) / 2 + P ⋅ P / ( 2 Y )
Y = 2 τ + a Q 1 + W ⋅ W / 2 , P = W Y , Z = 2 Y − τ . (33) Y=\frac{\sqrt{2}\tau+aQ}{1+W\cdot W/2},\qquad P=WY,\qquad Z=\sqrt{2}Y-\tau. \tag*{(33)} Y = 1 + W ⋅ W /2 2 τ + a Q , P = W Y , Z = 2 Y − τ . (33) The denominator is bounded away from zero on ∣ W ∣ ≤ w 0 |W|\leq w_0 ∣ W ∣ ≤ w 0 ∣ a Q ∣ ≤ s / 10 |aQ|\leq s/10 ∣ a Q ∣ ≤ s /10 y = ∣ Y ∣ ≍ s y=|Y|\asymp s y = ∣ Y ∣ ≍ s τ \tau τ Ω = { ( z , X , P , Z ) : a < s < 4 , ( X , P , Z ) ∈ Ω z } \Omega=\{(z,X,P,Z):a<s<4,\ (X,P,Z)\in\Omega_z\} Ω = {( z , X , P , Z ) : a < s < 4 , ( X , P , Z ) ∈ Ω z } H , G H,G H , G Lemma 3.1 . Their spatial parts at fixed z z z H f i b , G f i b H_{\mathrm{fib}},G_{\mathrm{fib}} H fib , G fib d ζ = d X d P d Z d\zeta=dX\,dP\,dZ d ζ = d X d P d Z
Lemma 4.1 (Fiber geometry). The spatial Hessian H f i b ( r ) H_{\mathrm{fib}}(r) H fib ( r ) H ( r ) H(r) H ( r ) G ( r ) ⊥ G(r)^\perp G ( r ) ⊥ ker H ( ∣ Φ ∣ 2 ) \ker H(|\Phi|^2) ker H ( ∣Φ ∣ 2 ) a < s < 4 a<s<4 a < s < 4
H f i b ( r ) ≥ c s − 2 I , G f i b ( r ) G f i b ( r ) ∗ ≤ H f i b ( r ) , (34) H_{\mathrm{fib}}(r)\geq cs^{-2}I,\qquad G_{\mathrm{fib}}(r)G_{\mathrm{fib}}(r)^*\leq H_{\mathrm{fib}}(r), \tag*{(34)} H fib ( r ) ≥ c s − 2 I , G fib ( r ) G fib ( r ) ∗ ≤ H fib ( r ) , (34) d P d Z ≍ a 2 s 2 d d W d Q , det H f i b ( r ) ≍ a − 2 s − 2 d . (35) dP\,dZ\asymp a^2s^{2d}\,dW\,dQ,\qquad\det H_{\mathrm{fib}}(r)\asymp a^{-2}s^{-2d}. \tag*{(35)} d P d Z ≍ a 2 s 2 d d W d Q , det H fib ( r ) ≍ a − 2 s − 2 d . (35) The spatial H f i b ( ∣ Φ ∣ 2 ) H_{\mathrm{fib}}(|\Phi|^2) H fib ( ∣Φ ∣ 2 ) ∣ G ( ∣ W ∣ 2 ) ∣ ≲ s − 1 |G(|W|^2)|\lesssim s^{-1} ∣ G ( ∣ W ∣ 2 ) ∣ ≲ s − 1
Proof. Set F = ( W / w 0 , 10 Q ) F=(W/w_0,10Q) F = ( W / w 0 , 10 Q ) J J J ( τ , P , Z ) (\tau,P,Z) ( τ , P , Z ) J f i b J_{\mathrm{fib}} J fib r = ∣ F ∣ 2 − 1 r=|F|^2-1 r = ∣ F ∣ 2 − 1 Lemma 3.1 gives
H ( r ) = 4 J ∗ J , G ( r ) = 2 J ∗ F . H(r)=4J^*J,\qquad G(r)=2J^*F. H ( r ) = 4 J ∗ J , G ( r ) = 2 J ∗ F . At fixed τ \tau τ J f i b J_{\mathrm{fib}} J fib τ \tau τ ( W , Q ) (W,Q) ( W , Q ) ker J \ker J ker J C k \mathbb{C}_k C k H ( r ) H(r) H ( r ) G ( r ) G(r) G ( r ) J ∗ J^* J ∗ ker H ( ∣ Φ ∣ 2 ) \ker H(|\Phi|^2) ker H ( ∣Φ ∣ 2 ) J J J ∂ Φ \partial\Phi ∂ Φ (33) , derivatives of ( P , Z ) (P,Z) ( P , Z ) W W W O ( s ) O(s) O ( s ) Q Q Q O ( a ) O(a) O ( a ) ∥ J f i b − 1 ∥ ≲ s \|J_{\mathrm{fib}}^{-1}\|\lesssim s ∥ J fib − 1 ∥ ≲ s ∣ F ∣ ≤ 1 |F|\leq1 ∣ F ∣ ≤ 1 (33) in ( W , Q ) (W,Q) ( W , Q ) 2 a Y d / ( 1 + W ⋅ W / 2 ) \sqrt{2}aY^d/(1+W\cdot W/2) 2 a Y d / ( 1 + W ⋅ W /2 ) P = W Y P=WY P = W Y d Y dY d Y Y d Y^d Y d W W W d Y dY d Y 4 J f i b ∗ J f i b 4J_{\mathrm{fib}}^*J_{\mathrm{fib}} 4 J fib ∗ J fib 4 n ∣ det C J f i b ∣ 2 4^n|\det_{\mathbb{C}}J_{\mathrm{fib}}|^2 4 n ∣ det C J fib ∣ 2 (35) . Finally the spatial derivatives of Φ \Phi Φ ∂ Z Φ = ( 1 − W ⋅ W / 2 ) / 2 \partial_Z\Phi=(1-W\cdot W/2)/\sqrt{2} ∂ Z Φ = ( 1 − W ⋅ W /2 ) / 2 W = P / Y W=P/Y W = P / Y O ( s − 1 ) O(s^{-1}) O ( s − 1 )
The Dirichlet potential
The time density can vanish. We first smooth the wave current in X X X Q Q Q H ( r ) H(r) H ( r ) ℓ > 0 \ell>0 ℓ > 0 J ℓ J_\ell J ℓ c n ℓ − n ( 1 + ∣ X / ℓ ∣ 2 ) − n − 2 c_n\ell^{-n}(1+|X/\ell|^2)^{-n-2} c n ℓ − n ( 1 + ∣ X / ℓ ∣ 2 ) − n − 2 0 ≤ g ≤ 1 0\leq g\leq1 0 ≤ g ≤ 1 C k \mathbb{C}^k C k ∣ Q ∣ ≤ 1 / 100 |Q|\leq1/100 ∣ Q ∣ ≤ 1/100 ∣ Q ∣ ≥ 1 / 20 |Q|\geq1/20 ∣ Q ∣ ≥ 1/20
A w = g ( Q ) ( J h ∗ X A 0 ) , A = A w + h 50 n C ( H ( r ) ) , μ = A T T . (36) A_w=g(Q)(J_h\ast_X A^0),\qquad A=A_w+h^{50n}C(H(r)),\qquad\mu=ATT. \tag*{(36)} A w = g ( Q ) ( J h ∗ X A 0 ) , A = A w + h 50 n C ( H ( r )) , μ = A T T . (36) Convolution takes place only in the real spatial variable. Since ∂ Φ \partial\Phi ∂ Φ X X X H ( ∣ Φ ∣ 2 ) H(|\Phi|^2) H ( ∣Φ ∣ 2 ) G ( g ( Q ) ) G(g(Q)) G ( g ( Q )) ker H ( r ) \ker H(r) ker H ( r ) G ( r ) G(r) G ( r ) T T TT T T det H f i b ( r ) > 0 \det H_{\mathrm{fib}}(r)>0 det H fib ( r ) > 0
A ≥ 0 , μ > 0 , A : H ( ∣ Φ ∣ 2 ) = 0. (37) A\geq0,\qquad\mu>0,\qquad A:H(|\Phi|^2)=0. \tag*{(37)} A ≥ 0 , μ > 0 , A : H ( ∣Φ ∣ 2 ) = 0. (37) Put B 0 = λ ∣ Φ ∣ 2 B_0=\lambda|\Phi|^2 B 0 = λ ∣Φ ∣ 2 B = H ( B 0 ) B=H(B_0) B = H ( B 0 ) B f i b = H f i b ( B 0 ) B_{\mathrm{fib}}=H_{\mathrm{fib}}(B_0) B fib = H fib ( B 0 ) λ = h − 30 n \lambda=h^{-30n} λ = h − 30 n (37) , the full background satisfies A : B = 0 A:B=0 A : B = 0 B f i b B_{\mathrm{fib}} B fib β ≍ λ \beta\asymp\lambda β ≍ λ h 50 n h^{50n} h 50 n λ \lambda λ
μ 1 / d ≤ λ (38) \mu^{1/d}\leq\lambda \tag*{(38)} μ 1/ d ≤ λ (38) for all sufficiently small h h h χ \chi χ ( κ / y ) d / 2 ≲ h − d (\kappa/y)^{d/2}\lesssim h^{-d} ( κ / y ) d /2 ≲ h − d V v V^v V v ∑ ∣ c κ , v ∣ ≲ h − n / 2 \sum|c_{\kappa,v}|\lesssim h^{-n/2} ∑ ∣ c κ , v ∣ ≲ h − n /2 ∥ A 0 ∥ ≲ h − n − 2 d \|A^0\|\lesssim h^{-n-2d} ∥ A 0 ∥ ≲ h − n − 2 d O ( h 50 n a − 2 s − 2 d ) O(h^{50n}a^{-2}s^{-2d}) O ( h 50 n a − 2 s − 2 d ) μ \mu μ (35) .
On each spatial fiber there is a unique smooth real u u u
D u = H f i b ( B 0 + u ) > 0 , det D u = λ μ , u ∣ ∂ Ω z = 0. (39) D_u=H_{\mathrm{fib}}(B_0+u)>0,\qquad\det D_u=\lambda\mu,\qquad u|_{\partial\Omega_z}=0. \tag*{(39)} D u = H fib ( B 0 + u ) > 0 , det D u = λ μ , u ∣ ∂ Ω z = 0. (39) Indeed, the spatial defining Hessian is positive definite, and B f i b ≥ 0 B_{\mathrm{fib}}\geq0 B fib ≥ 0 n ≥ 2 n\geq2 n ≥ 2 μ > 0 \mu>0 μ > 0 (33) and the parameter assertion of that theorem give smooth dependence on z z z u ≤ 0 u\leq0 u ≤ 0 α I \alpha I α I det ( B f i b + α I ) = ( β + α ) α d \det(B_{\mathrm{fib}}+\alpha I)=(\beta+\alpha)\alpha^d det ( B fib + α I ) = ( β + α ) α d 0 < α ≤ λ 0<\alpha\leq\lambda 0 < α ≤ λ λ μ \lambda\mu λ μ α ≍ μ 1 / d \alpha\asymp\mu^{1/d} α ≍ μ 1/ d
Cancelling the full determinant
The quantity that measures the depth of the potential against its density is X 0 ( z ) = ∫ Ω z ( − u ) μ d ζ X_0(z)=\int_{\Omega_z}(-u)\,\mu\,d\zeta X 0 ( z ) = ∫ Ω z ( − u ) μ d ζ A A A Δ z X 0 \Delta_zX_0 Δ z X 0 − ∫ Ω z A : H ( u ) d ζ -\int_{\Omega_z}A:H(u)\,d\zeta − ∫ Ω z A : H ( u ) d ζ [71 Proposition 2.2] . Here only the spatial block of H ( B 0 + u ) H(B_0+u) H ( B 0 + u )
H ( B 0 + u ) = ( ℓ c ∗ c D u ) = σ 0 e T e T ∗ + Σ , σ 0 = ℓ − c ∗ D u − 1 c , Σ = ( c ∗ D u − 1 c c ∗ c D u ) ≥ 0. (40) H(B_0+u)=\begin{pmatrix}\ell& c^* \\ c & D_u\end{pmatrix}=\sigma_0 e_T e_T^*+\Sigma,\qquad\sigma_0=\ell-c^*D_u^{-1}c,\qquad\Sigma=\begin{pmatrix}c^*D_u^{-1}c & c^* \\ c & D_u\end{pmatrix}\geq0. \tag*{(40)} H ( B 0 + u ) = ( ℓ c c ∗ D u ) = σ 0 e T e T ∗ + Σ , σ 0 = ℓ − c ∗ D u − 1 c , Σ = ( c ∗ D u − 1 c c c ∗ D u ) ≥ 0. (40) Here e T e_T e T Σ ≥ 0 \Sigma\geq0 Σ ≥ 0 A A A A : B A:B A : B (37) ; in the determinant, det D u = λ μ \det D_u=\lambda\mu det D u = λ μ det H ( B 0 + u ) = λ μ σ 0 \det H(B_0+u)=\lambda\mu\sigma_0 det H ( B 0 + u ) = λ μ σ 0
A : H ( u ) = A : Σ + μ σ 0 = A : Σ + λ − 1 det H ( B 0 + u ) . (41) A:H(u)=A:\Sigma+\mu\sigma_0=A:\Sigma+\lambda^{-1}\det H(B_0+u). \tag*{(41)} A : H ( u ) = A : Σ + μ σ 0 = A : Σ + λ − 1 det H ( B 0 + u ) . (41) The full determinant has no prescribed sign. We look for a second fiber integral whose Laplacian cancels this term. Write D v = H f i b ( B 0 + v ) D_v=H_{\mathrm{fib}}(B_0+v) D v = H fib ( B 0 + v ) B 0 + θ u B_0+\theta u B 0 + θ u T T TT T T det D θ u \det D_{\theta u} det D θ u H ( u ) H(u) H ( u ) θ \theta θ v v v Ω z \Omega_z Ω z D v > 0 D_v>0 D v > 0
E z ( v ) = 1 λ ∫ Ω z v ∫ 0 1 det D θ v d θ d ζ , J z ( v ) = E z ( v ) − ∫ Ω z v μ d ζ . (42) E_z(v)=\frac{1}{\lambda}\int_{\Omega_z}v\int_0^1\det D_{\theta v}\,d\theta\,d\zeta,\qquad\mathcal{J}_z(v)=E_z(v)-\int_{\Omega_z}v\mu\,d\zeta. \tag*{(42)} E z ( v ) = λ 1 ∫ Ω z v ∫ 0 1 det D θ v d θ d ζ , J z ( v ) = E z ( v ) − ∫ Ω z v μ d ζ . (42) The segment integral is the quaternionic counterpart of the Aubin–Mabuchi energy: Mabuchi’s functional L L L [52 (2.2.1)] . Here the determinant is a polynomial, so the integrand is smooth up to the singular endpoint D 0 = B f i b D_0=B_{\mathrm{fib}} D 0 = B fib F ( z ) = J z ( u ) = E z ( u ) + X 0 ( z ) \mathcal{F}(z)=\mathcal{J}_z(u)=E_z(u)+X_0(z) F ( z ) = J z ( u ) = E z ( u ) + X 0 ( z )
Proposition 4.2 (Energy balance). The energy F \mathcal{F} F a < ∣ τ ∣ < 4 a<|\tau|<4 a < ∣ τ ∣ < 4 ρ \rho ρ u / r u/r u / r z = ( T , τ ) z=(T,\tau) z = ( T , τ )
Δ z F ( z ) + ∫ Ω z A : Σ d ζ = ∫ ∂ Ω z ρ G ( r ) ∗ A w G ( r ) δ ( r ) d ζ . (43) \Delta_z\mathcal{F}(z)+\int_{\Omega_z}A:\Sigma\,d\zeta=\int_{\partial\Omega_z}\rho\,G(r)^*A_{\mathrm{w}}G(r)\,\delta(r)\,d\zeta. \tag*{(43)} Δ z F ( z ) + ∫ Ω z A : Σ d ζ = ∫ ∂ Ω z ρ G ( r ) ∗ A w G ( r ) δ ( r ) d ζ . (43) The normal seen by the current is that of the moving wall in the total space, so G ( r ) G(r) G ( r ) δ ( r ) d ζ = d S f i b / ∣ ∇ f i b r ∣ \delta(r)\,d\zeta=dS_{\mathrm{fib}}/|\nabla_{\mathrm{fib}}r| δ ( r ) d ζ = d S fib /∣ ∇ fib r ∣ A : Σ ≥ 0 A:\Sigma\geq0 A : Σ ≥ 0 Δ z F \Delta_z\mathcal{F} Δ z F
Proof. Denote the boundary integral in (43) by B ( z ) B(z) B ( z )
Δ z X 0 = B ( z ) − ∫ Ω z A : Σ d ζ − λ − 1 ∫ Ω z det H ( B 0 + u ) d ζ , \Delta_zX_0=B(z)-\int_{\Omega_z}A:\Sigma\,d\zeta-\lambda^{-1}\int_{\Omega_z}\det H(B_0+u)\,d\zeta, Δ z X 0 = B ( z ) − ∫ Ω z A : Σ d ζ − λ − 1 ∫ Ω z det H ( B 0 + u ) d ζ , Δ z [ E z ( u ) ] = λ − 1 ∫ Ω z det H ( B 0 + u ) d ζ . \Delta_z[E_z(u)]=\lambda^{-1}\int_{\Omega_z}\det H(B_0+u)\,d\zeta. Δ z [ E z ( u )] = λ − 1 ∫ Ω z det H ( B 0 + u ) d ζ . For a fixed complex product domain, Åhag et al. [1 Theorem 3.1] identified the Laplacian of the slice energy, up to normalization, with the integral of the full Monge–Ampère measure over the fiber. Here the fiber moves, so the second identity requires a boundary calculation. Both follow by testing with a smooth η = η ( z ) \eta=\eta(z) η = η ( z ) H ( η ) H(\eta) H ( η ) H T T ( η ) = Δ z η H_{TT}(\eta)=\Delta_z\eta H T T ( η ) = Δ z η M M M (30) on the total domain gives
∫ Ω η M T T Δ z η = ∫ Ω η M : H ( u ) − ∫ ∂ l a t Ω η ( ν ∗ M ν ) ∂ n u d S . (44) \int_{\Omega}\eta M_{TT}\Delta_z\eta=\int_{\Omega}\eta M:H(u)-\int_{\partial_{\mathrm{lat}}\Omega}\eta(\nu^*M\nu)\,\partial_nu\,dS. \tag*{(44)} ∫ Ω η M T T Δ z η = ∫ Ω η M : H ( u ) − ∫ ∂ lat Ω η ( ν ∗ M ν ) ∂ n u d S . (44) Here ∂ l a t Ω \partial_{\mathrm{lat}}\Omega ∂ lat Ω r = 0 r=0 r = 0 Ω \Omega Ω d z d ζ dz\,d\zeta d z d ζ u ∇ η u\nabla\eta u ∇ η u = 0 u=0 u = 0 η \eta η n = ∇ r / ∣ ∇ r ∣ \mathbf{n}=\nabla r/\lvert\nabla r\rvert n = ∇ r / ∣ ∇ r ∣ ν = G ( r ) / ∣ ∇ r ∣ \nu=G(r)/\lvert\nabla r\rvert ν = G ( r ) / ∣ ∇ r ∣ r r r u = r ρ u=r\rho u = r ρ ρ \rho ρ ρ ≥ 0 \rho\ge0 ρ ≥ 0 r = 0 r=0 r = 0 ∂ n u = ρ ∣ ∇ r ∣ \partial_{\mathbf{n}}u=\rho\lvert\nabla r\rvert ∂ n u = ρ ∣ ∇ r ∣
For X 0 X_0 X 0 M = A M=A M = A (44) and change sign since X 0 X_0 X 0 ( − u ) μ (-u)\mu ( − u ) μ (41) . On the wall the regularizer C ( H ( r ) ) C(H(r)) C ( H ( r )) ν \nu ν A w A_w A w ∂ n u = ρ ∣ ∇ r ∣ \partial_{\mathbf{n}}u=\rho\lvert\nabla r\rvert ∂ n u = ρ ∣ ∇ r ∣ d S = ∣ ∇ r ∣ δ ( r ) d z d ζ dS=\lvert\nabla r\rvert\delta(r)\,dz\,d\zeta d S = ∣ ∇ r ∣ δ ( r ) d z d ζ B ( z ) B(z) B ( z )
For E z ( u ) E_z(u) E z ( u ) M = C ( H ( B 0 + θ u ) ) M=C(H(B_0+\theta u)) M = C ( H ( B 0 + θ u )) (31) gives, on r = 0 r=0 r = 0
H ( B 0 + θ u ) ∣ ν ⊥ = ( B + θ ρ H ( r ) ) ∣ ν ⊥ . H(B_0+\theta u)|_{\nu^\perp}=(B+\theta\rho H(r))|_{\nu^\perp}. H ( B 0 + θ u ) ∣ ν ⊥ = ( B + θ ρ H ( r )) ∣ ν ⊥ . The null line of H ( r ) H(r) H ( r ) ν ⊥ \nu^\perp ν ⊥ ker B \ker B ker B ν ∗ M ν = 0 \nu^*M\nu=0 ν ∗ M ν = 0 e T e_T e T M T T = det D θ u M_{TT}=\det D_\theta u M T T = det D θ u (44) in θ \theta θ λ \lambda λ ∫ E z ( u ) Δ z η d z \int E_z(u)\Delta_z\eta\,dz ∫ E z ( u ) Δ z η d z M : H ( u ) = ∂ θ det H ( B 0 + θ u ) M:H(u)=\partial_\theta\det H(B_0+\theta u) M : H ( u ) = ∂ θ det H ( B 0 + θ u ) θ = 1 \theta=1 θ = 1 det B = 0 \det B=0 det B = 0 (33) makes the volume and surface integrals smooth in z z z
Size of the energy on a fiber
Fix z z z (42) is a primitive of the spatial determinant: for a smooth variation w w w
d J z ( v ) [ w ] = ∫ Ω z w ( λ − 1 det D v − μ ) d ζ . dJ_z(v)[w]=\int_{\Omega_z}w\left(\lambda^{-1}\det D_v-\mu\right)\,d\zeta. d J z ( v ) [ w ] = ∫ Ω z w ( λ − 1 det D v − μ ) d ζ . d 2 J z ( v ) [ w , w ] = − λ − 1 ∫ Ω z G f i b ( w ) ∗ C ( D v ) G f i b ( w ) d ζ . d^2J_z(v)[w,w]=-\lambda^{-1}\int_{\Omega_z}G_{\mathrm{fib}}(w)^*C(D_v)G_{\mathrm{fib}}(w)\,d\zeta. d 2 J z ( v ) [ w , w ] = − λ − 1 ∫ Ω z G fib ( w ) ∗ C ( D v ) G fib ( w ) d ζ . For the first identity, differentiating under the integral in (42) produces a term θ v C ( D θ v ) : H f i b ( w ) \theta vC(D_{\theta v}):H_{\mathrm{fib}}(w) θ v C ( D θ v ) : H fib ( w ) v v v w w w θ \theta θ w ∂ θ ( θ det D θ v ) w\,\partial_\theta(\theta\det D_{\theta v}) w ∂ θ ( θ det D θ v ) w w w C ( D v ) > 0 C(D_v)>0 C ( D v ) > 0 [76 Section 4] .
Since u ≤ 0 u\le0 u ≤ 0 D θ u = ( 1 − θ ) B f i b + θ D u ≥ 0 D_{\theta u}=(1-\theta)B_{\mathrm{fib}}+\theta Du\ge0 D θ u = ( 1 − θ ) B fib + θ D u ≥ 0 E z ( u ) E_z(u) E z ( u ) F \mathcal{F} F B 0 + u B_0+u B 0 + u λ F \lambda\mathcal{F} λ F J \mathcal{J} J [5 p. 145] , with B 0 + u B_0+u B 0 + u B 0 B_0 B 0 [10 Section 2.1] . The rank-one background lets us calculate how much of X 0 X_0 X 0 F = X 0 + E z ( u ) \mathcal{F}=X_0+E_z(u) F = X 0 + E z ( u )
Lemma 4.3 (Energy comparison). On every spatial fiber,
F = n − 1 n + 1 X 0 + 1 λ n ( n + 1 ) ∫ Ω z G f i b ( u ) ∗ C ( D u ) G f i b ( u ) d ζ . (45) \mathcal{F}=\frac{n-1}{n+1}X_0+\frac{1}{\lambda n(n+1)}\int_{\Omega_z}G_{\mathrm{fib}}(u)^*C(D_u)G_{\mathrm{fib}}(u)\,d\zeta. \tag*{(45)} F = n + 1 n − 1 X 0 + λn ( n + 1 ) 1 ∫ Ω z G fib ( u ) ∗ C ( D u ) G fib ( u ) d ζ . (45) In particular, n − 1 n + 1 X 0 ≤ F ≤ n n + 1 X 0 \frac{n-1}{n+1}X_0\le\mathcal{F}\le\frac{n}{n+1}X_0 n + 1 n − 1 X 0 ≤ F ≤ n + 1 n X 0
Proof. Because B f i b B_{\mathrm{fib}} B fib
det D θ u = θ n det D u + ( 1 − θ ) θ n − 1 C ( D u ) : B f i b . \det D_{\theta}u=\theta^{n}\det Du+(1-\theta)\theta^{n-1}C(Du):B_{\mathrm{fib}}. det D θ u = θ n det D u + ( 1 − θ ) θ n − 1 C ( D u ) : B fib . Write I = ∫ Ω ζ ( − u ) C ( D u ) : B f i b d ζ ≥ 0 I=\int_{\Omega_{\zeta}}(-u)C(Du):B_{\mathrm{fib}}\,d\zeta\ge0 I = ∫ Ω ζ ( − u ) C ( D u ) : B fib d ζ ≥ 0 θ \theta θ F = n X 0 / ( n + 1 ) − I / [ λ n ( n + 1 ) ] \mathcal{F}=nX_{0}/(n+1)-I/[\lambda^{n}(n+1)] F = n X 0 / ( n + 1 ) − I / [ λ n ( n + 1 )] C ( D u ) : D u = n det D u C(Du):Du=n\det Du C ( D u ) : D u = n det D u
n λ X 0 − I = ∫ Ω ζ ( − u ) C ( D u ) : H f i b ( u ) d ζ = ∫ Ω ζ G f i b ( u ) ∗ C ( D u ) G f i b ( u ) d ζ . n\lambda X_{0}-I=\int_{\Omega_{\zeta}}(-u)C(Du):H_{\mathrm{fib}}(u)\,d\zeta=\int_{\Omega_{\zeta}}G_{\mathrm{fib}}(u)^{*}C(Du)G_{\mathrm{fib}}(u)\,d\zeta. nλ X 0 − I = ∫ Ω ζ ( − u ) C ( D u ) : H fib ( u ) d ζ = ∫ Ω ζ G fib ( u ) ∗ C ( D u ) G fib ( u ) d ζ . This last integration by parts uses u = 0 u=0 u = 0 (45) and its lower bound.
Estimating the energy on a fixed fiber
On a fiber at fixed ( T , τ ) (T,\tau) ( T , τ ) r = 0 r=0 r = 0 u / r u/r u / r u u u h h h
Proposition 5.1. Fix an annulus c 1 ≤ h ∣ ξ ∣ ≤ c 2 c_{1}\le h|\xi|\le c_{2} c 1 ≤ h ∣ ξ ∣ ≤ c 2 0 < c 1 < c 2 0<c_{1}<c_{2} 0 < c 1 < c 2 γ > 0 \gamma>0 γ > 0 K ≥ 1 K\ge1 K ≥ 1 c , C > 0 c,C>0 c , C > 0 n n n a = K h a=Kh a = K h 0 < h < 1 / ( 4 K ) 0<h<1/(4K) 0 < h < 1/ ( 4 K ) f f f ∥ f ∥ 2 = 1 \|f\|_{2}=1 ∥ f ∥ 2 = 1 ∣ v ∣ ≤ γ |v|\le\gamma ∣ v ∣ ≤ γ g T = e − i T ∣ D ∣ f g_{T}=e^{-iT|D|}f g T = e − i T ∣ D ∣ f F ( T , τ ) \mathcal{F}(T,\tau) F ( T , τ ) § 4 . Then
F ( 0 , τ ) ≤ C a 2 ∥ f ∥ p c 2 , ( ∣ τ ∣ = 4 a ) , (46) \mathcal{F}(0,\tau)\le Ca^{2}\|f\|_{p_{c}}^{2},\qquad(|\tau|=4a), \tag*{(46)} F ( 0 , τ ) ≤ C a 2 ∥ f ∥ p c 2 , ( ∣ τ ∣ = 4 a ) , (46) F ( T , τ ) ≥ c a 2 ∥ g T ∥ p c 2 , ( ∣ τ ∣ = 2 a ) , (47) \mathcal{F}(T,\tau)\ge ca^{2}\|g_{T}\|_{p_{c}}^{2},\qquad(|\tau|=2a), \tag*{(47)} F ( T , τ ) ≥ c a 2 ∥ g T ∥ p c 2 , ( ∣ τ ∣ = 2 a ) , (47) Δ z F ( T , τ ) ≤ V ( s ) F ( T , τ ) , V ( s ) = C s − 2 e − c s / h ( 2 a ≤ s = ∣ τ ∣ ≤ 3 ) . (48) \Delta_{z}\mathcal{F}(T,\tau)\le V(s)\mathcal{F}(T,\tau),\qquad V(s)=Cs^{-2}e^{-cs/h}\qquad(2a\le s=|\tau|\le3). \tag*{(48)} Δ z F ( T , τ ) ≤ V ( s ) F ( T , τ ) , V ( s ) = C s − 2 e − cs / h ( 2 a ≤ s = ∣ τ ∣ ≤ 3 ) . (48) The last two inequalities hold for all T ∈ R T\in\mathbb{R} T ∈ R
Proof. Write d = n − 1 d=n-1 d = n − 1 q = n / d = p c / 2 q=n/d=p_{c}/2 q = n / d = p c /2 σ = s / h \sigma=s/h σ = s / h 2 a ≤ s ≤ 3 2a\le s\le3 2 a ≤ s ≤ 3
r = ∣ W ∣ 2 w 0 2 + 100 ∣ Q ∣ 2 − 1 < 0 , w 0 = 1 / 10 , Q = Φ / a , r=\frac{|W|^{2}}{w_{0}^{2}}+100|Q|^{2}-1<0,\qquad w_{0}=1/10,\qquad Q=\Phi/a, r = w 0 2 ∣ W ∣ 2 + 100∣ Q ∣ 2 − 1 < 0 , w 0 = 1/10 , Q = Φ/ a , we use two regions. On r = 0 r=0 r = 0 ∣ Q ∣ ≤ 1 / 20 |Q|\le1/20 ∣ Q ∣ ≤ 1/20 ∣ W ∣ ≥ ( 3 / 2 ) w 0 |W|\ge(\sqrt{3}/2)w_{0} ∣ W ∣ ≥ ( 3 /2 ) w 0 ∣ W ∣ ≤ γ |W|\le\gamma ∣ W ∣ ≤ γ ∣ Q ∣ ≤ 1 / 400 |Q|\le1/400 ∣ Q ∣ ≤ 1/400 Figure 1 shows the separation between these regions. It lets us compare two exponentials with different constants in their exponents. For now take 0 < γ < w 0 / 8 0<\gamma<w_{0}/8 0 < γ < w 0 /8 s s s γ \gamma γ
Figure 1. The fixed fiber in modulus coordinates. The wave flux through r = 0 r = 0 r = 0 ∣ Q ∣ ≤ 1 / 20 |Q| \le1/20 ∣ Q ∣ ≤ 1/20 ∣ W ∣ ≥ 3 w 0 / 2 |W| \ge\sqrt{3}w_0/2 ∣ W ∣ ≥ 3 w 0 /2 e − c s / h e^{-cs/h} e − cs / h ∣ v ∣ ≤ w 0 / 3 |v| \le w_0/3 ∣ v ∣ ≤ w 0 /3 K γ K\gamma K γ h h h ∣ W − v ∣ ≤ 4 γ |W-v| \le4\gamma ∣ W − v ∣ ≤ 4 γ ∣ v ∣ ≤ 2 γ |v| \le2\gamma ∣ v ∣ ≤ 2 γ e C γ 2 s / h e^{C\gamma^2s/h} e C γ 2 s / h γ \gamma γ h h h w 0 w_0 w 0
A spatial average for the upper bound. For a dimensional constant D ≥ 1 D\ge1 D ≥ 1
P ∗ ( X ) = ( ( J D s ∗ ∣ g T ∣ 2 ) ( X ) ) 1 / d , M = 2 n + 4. P_{*}(X)=\left((J_{Ds}*|g_{T}|^{2})(X)\right)^{1/d},\qquad M=2n+4. P ∗ ( X ) = ( ( J D s ∗ ∣ g T ∣ 2 ) ( X ) ) 1/ d , M = 2 n + 4. Here and below convolution is in the real spatial variables X X X J ℓ J_{\ell} J ℓ C n ℓ − n ( 1 + ∣ X / ℓ ∣ 2 ) − M / 2 C_{n}\ell^{-n}(1+|X/\ell|^{2})^{-M/2} C n ℓ − n ( 1 + ∣ X / ℓ ∣ 2 ) − M /2 d d d
∣ ∇ X j P ∗ ∣ ≤ C ( D s ) − j P ∗ ( j = 1 , 2 ) , ∫ P ∗ d + 1 d X ≤ ∥ g T ∥ p c p c . (49) |\nabla_{X}^{j}P_{*}|\le C(Ds)^{-j}P_{*}\quad(j=1,2),\qquad\int P_{*}^{d+1}\,dX\le\|g_{T}\|_{p_{c}}^{p_{c}}. \tag*{(49)} ∣ ∇ X j P ∗ ∣ ≤ C ( D s ) − j P ∗ ( j = 1 , 2 ) , ∫ P ∗ d + 1 d X ≤ ∥ g T ∥ p c p c . (49) The integral bound is Young’s inequality with exponent q q q P ∗ > 0 P_* > 0 P ∗ > 0 ∥ g T ∥ 2 2 = 1 \|g_T\|_2^2 = 1 ∥ g T ∥ 2 2 = 1
We next bound the wave current A w A_w A w μ \mu μ g T g_T g T
( κ y ) d / 2 exp ( − Ψ ) , Ψ = 1 2 κ y ∣ W − v ∣ 2 , y = ∣ Y ∣ ≍ s . \left(\frac{\kappa}{y}\right)^{d/2}\exp(-\Psi), \qquad\Psi= \frac{1}{2}\kappa y|W-v|^2, \qquad y = |Y| \asymp s. ( y κ ) d /2 exp ( − Ψ ) , Ψ = 2 1 κ y ∣ W − v ∣ 2 , y = ∣ Y ∣ ≍ s . Multiply the symbol by a fixed smooth annular cutoff equal to one on the data and supported in ∣ v ∣ < w 0 / 3 |v| < w_0/3 ∣ v ∣ < w 0 /3 η = h ξ \eta= h\xi η = h ξ M M M C ( h s ) − d / 2 σ m C(hs)^{-d/2}\sigma^m C ( h s ) − d /2 σ m m m m h κ h\kappa hκ v v v h κ h\kappa hκ Ψ \Psi Ψ C σ C\sigma C σ r = 0 r = 0 r = 0 g ( Q ) g(Q) g ( Q ) ∣ W − v ∣ |W-v| ∣ W − v ∣ e − c 0 σ e^{-c_0\sigma} e − c 0 σ c 0 > 0 c_0 > 0 c 0 > 0 γ \gamma γ
If B B B ( 1 − Δ η ) n + 2 (1-\Delta_\eta)^{n+2} ( 1 − Δ η ) n + 2 C B h − n ( 1 + ∣ X / h ∣ ) − M / 2 CBh^{-n}(1+|X/h|)^{-M/2} C B h − n ( 1 + ∣ X / h ∣ ) − M /2 C B J h CBJ_h C B J h C B 2 J h ∗ ∣ g T ∣ 2 CB^2J_h * |g_T|^2 C B 2 J h ∗ ∣ g T ∣ 2 J h J_h J h A w A_w A w J h ∗ J h ≤ C J h J_h * J_h \le CJ_h J h ∗ J h ≤ C J h ∣ Y ∣ |Y| ∣ Y ∣ ∣ X − Y ∣ |X-Y| ∣ X − Y ∣ ∣ X ∣ / 2 |X|/2 ∣ X ∣/2 J h ≤ ( D s / h ) n J D s J_h \le(D s/h)^n J_{Ds} J h ≤ ( D s / h ) n J D s ( h s ) − d = s − 2 d σ d (hs)^{-d} = s^{-2d}\sigma^d ( h s ) − d = s − 2 d σ d m m m
∥ A w ∥ ≤ C s − 2 d σ m P ∗ d ( r ≤ 0 ) , \|A_w\| \le Cs^{-2d}\sigma^m P_*^d \qquad(r \le0), ∥ A w ∥ ≤ C s − 2 d σ m P ∗ d ( r ≤ 0 ) , ∥ A w ∥ ≤ C s − 2 d σ m e − c 0 σ P ∗ d ( r = 0 ) . (50) \|A_w\| \le Cs^{-2d}\sigma^m e^{-c_0\sigma}P_*^d \qquad(r = 0). \tag*{(50)} ∥ A w ∥ ≤ C s − 2 d σ m e − c 0 σ P ∗ d ( r = 0 ) . (50) Here ∥ A w ∥ \|A_w\| ∥ A w ∥ μ \mu μ h 50 n det H f i b ( r ) ≤ C h 50 n a − 2 s − 2 d h^{50n}\det H_{\mathrm{fib}}(r) \le Ch^{50n}a^{-2}s^{-2d} h 50 n det H fib ( r ) ≤ C h 50 n a − 2 s − 2 d Lemma 4.1 . The nearest lattice term in the positive kernel gives inf X J D s ( X ) ≥ c D h M − n \inf_X J_{Ds}(X) \ge c_Dh^{M-n} inf X J D s ( X ) ≥ c D h M − n h ≤ s ≤ 3 h \le s \le3 h ≤ s ≤ 3 P ∗ d ≥ c D h M − n ∥ g T ∥ 2 2 = c D h M − n P_*^d \ge c_Dh^{M-n}\|g_T\|_2^2 = c_Dh^{M-n} P ∗ d ≥ c D h M − n ∥ g T ∥ 2 2 = c D h M − n h 50 n a − 2 h^{50n}a^{-2} h 50 n a − 2
μ ≤ C s − 2 d σ m P ∗ d ( r ≤ 0 ) . (51) \mu\le Cs^{-2d}\sigma^mP_*^d \qquad(r \le0). \tag*{(51)} μ ≤ C s − 2 d σ m P ∗ d ( r ≤ 0 ) . (51) A comparison function that vanishes on the wall. Put R = H f i b ( r ) R=H_{\mathrm{fib}}(r) R = H fib ( r ) R ≥ c s − 2 I R\ge cs^{-2}I R ≥ c s − 2 I G f i b ( r ) G f i b ( r ) ∗ ≤ C R G_{\mathrm{fib}}(r)G_{\mathrm{fib}}(r)^{*}\le CR G fib ( r ) G fib ( r ) ∗ ≤ C R
H f i b ( r P ∗ ) ≥ ( 1 − C / D ) P ∗ R . (52) H_{\mathrm{fib}}(rP_{*})\ge(1-C/D)P_{*}R. \tag*{(52)} H fib ( r P ∗ ) ≥ ( 1 − C / D ) P ∗ R . (52) In fact, the product rule of Lemma 3.1 gives P ∗ R P_{*}R P ∗ R r H f i b ( P ∗ ) rH_{\mathrm{fib}}(P_{*}) r H fib ( P ∗ ) r r r P ∗ P_{*} P ∗ e e e e e e C ( e ∗ R e ) 1 / 2 C(e^{*}Re)^{1/2} C ( e ∗ R e ) 1/2 C D − 1 P ∗ ( e ∗ R e ) 1 / 2 CD^{-1}P_{*}(e^{*}Re)^{1/2} C D − 1 P ∗ ( e ∗ R e ) 1/2 H f i b ( P ∗ ) H_{\mathrm{fib}}(P_{*}) H fib ( P ∗ ) C D − 2 CD^{-2} C D − 2 (49) and ∣ r ∣ ≤ 1 |r|\le1 ∣ r ∣ ≤ 1
Since r < 0 r<0 r < 0 u / r u/r u / r u u u λ μ \lambda\mu λ μ D D D (52) is at least P ∗ R / 2 P_{*}R/2 P ∗ R /2 B f i b = λ H f i b ( ∣ Φ ∣ 2 ) B_{\mathrm{fib}}=\lambda H_{\mathrm{fib}}(|\Phi|^{2}) B fib = λ H fib ( ∣Φ ∣ 2 ) λ = h − 30 n \lambda=h^{-30n} λ = h − 30 n v ∗ = C 1 σ m 1 r P ∗ v_{*}=C_{1}\sigma^{m_{1}}rP_{*} v ∗ = C 1 σ m 1 r P ∗
det ( B f i b + H f i b ( v ∗ ) ) ≥ c λ ( C 1 σ m 1 P ∗ s − 2 ) d ≥ λ μ \det\left(B_{\mathrm{fib}}+H_{\mathrm{fib}}(v_{*})\right)\ge c\lambda\left(C_{1}\sigma^{m_{1}}P_{*}s^{-2}\right)^{d}\ge\lambda\mu det ( B fib + H fib ( v ∗ ) ) ≥ c λ ( C 1 σ m 1 P ∗ s − 2 ) d ≥ λ μ by (51) with m 1 , C 1 m_{1},C_{1} m 1 , C 1 v ∗ v_{*} v ∗ u u u Theorem A.1 ) yields v ∗ ≤ u ≤ 0 v_{*}\le u\le0 v ∗ ≤ u ≤ 0 r < 0 r<0 r < 0
0 ≤ u / r ≤ C 1 σ m 1 P ∗ ( r ≤ 0 ) . (53) 0\le u/r\le C_{1}\sigma^{m_{1}}P_{*}\qquad(r\le0). \tag*{(53)} 0 ≤ u / r ≤ C 1 σ m 1 P ∗ ( r ≤ 0 ) . (53) By Lemma 4.3, F ≤ R ( − u ) μ \mathcal{F}\le R(-u)^{\mu} F ≤ R ( − u ) μ a 2 s 2 d d W d Q a^{2}s^{2d}\,dW\,dQ a 2 s 2 d d W d Q (49) , (51) and (53) gives
F ( T , τ ) ≤ C a 2 σ m ∥ g T ∥ p c p c . (54) \mathcal{F}(T,\tau)\le Ca^{2}\sigma^{m}\|g_{T}\|_{p^{c}}^{p^{c}}. \tag*{(54)} F ( T , τ ) ≤ C a 2 σ m ∥ g T ∥ p c p c . (54) For the flux we need the full gradient of r r r A W G ( ∣ Φ ∣ 2 ) = 0 A_{W}G(|\Phi|^{2})=0 A W G ( ∣Φ ∣ 2 ) = 0 G ( ∣ W ∣ 2 / ω 0 2 ) G(|W|^{2}/\omega_{0}^{2}) G ( ∣ W ∣ 2 / ω 0 2 ) O ( s − 1 ) O(s^{-1}) O ( s − 1 ) W = P / Y W=P/Y W = P / Y y ≍ s y\asymp s y ≍ s G ( r ) ∗ A W G ( r ) ≤ C s − 2 ∥ A W ∥ G(r)^{*}A_{W}G(r)\le Cs^{-2}\|A_{W}\| G ( r ) ∗ A W G ( r ) ≤ C s − 2 ∥ A W ∥ ∫ δ ( r ) d P d Z ≤ C a 2 s 2 d \int\delta(r)\,dP\,dZ\le Ca^{2}s^{2d} ∫ δ ( r ) d P d Z ≤ C a 2 s 2 d ( W , Q ) (W,Q) ( W , Q ) (50) and (53) to the flux integral in Proposition 4.2, we conclude that
Δ z F ( T , τ ) ≤ C ( a / s ) 2 σ m e − c 0 σ ∥ g T ∥ p c p c . (55) \Delta_{z}\mathcal{F}(T,\tau)\le C(a/s)^{2}\sigma^{m}e^{-c_{0}\sigma}\|g_{T}\|_{p^{c}}^{p^{c}}. \tag*{(55)} Δ z F ( T , τ ) ≤ C ( a / s ) 2 σ m e − c 0 σ ∥ g T ∥ p c p c . (55) Recovering the wave in the interior. Consider a center ( X , P , Z ) (X,P,Z) ( X , P , Z ) ∣ W ∣ ≤ γ |W|\le\gamma ∣ W ∣ ≤ γ ∣ Q ∣ ≤ 1 / 400 |Q|\le1/400 ∣ Q ∣ ≤ 1/400 h h h 3 n 3n 3 n ∣ W ∣ ≤ 2 γ |W|\le2\gamma ∣ W ∣ ≤ 2 γ ∣ Q ∣ ≤ 1 / 100 |Q|\le1/100 ∣ Q ∣ ≤ 1/100 K γ K\gamma K γ K K K W = P / Y W=P/Y W = P / Y C h / s Ch/s C h / s Φ \Phi Φ C h Ch C h A W A_{W} A W
The first scalar multiplier S 0 S_{0} S 0 (17) has symbol b 0 ( κ / y ) d / 2 e − Ψ b_{0}(\kappa/y)^{d/2}e^{-\Psi} b 0 ( κ / y ) d /2 e − Ψ b 0 = ( 1 + Re ( W ) ⋅ v / 2 ) / 2 b_{0}=(1+\operatorname{Re}(W)\cdot v/2)/\sqrt{2} b 0 = ( 1 + Re ( W ) ⋅ v /2 ) / 2 ∣ S 0 g T ∣ 2 ≤ A T 0 T |S_{0}g_{T}|^{2}\le A_{T}^{0}{}_{T} ∣ S 0 g T ∣ 2 ≤ A T 0 T S 0 S_{0} S 0 g T g_{T} g T ∣ v ∣ ≤ 2 γ |v|\le2\gamma ∣ v ∣ ≤ 2 γ Ψ ≤ C γ 2 σ \Psi\le C\gamma^{2}\sigma Ψ ≤ C γ 2 σ η = h ξ \eta=h\xi η = h ξ M M M
B i n v = C γ ( h s ) d / 2 σ m e C γ 2 σ . B_{\mathrm{inv}}=C_{\gamma}(hs)^{d/2}\sigma^{m}e^{C\gamma^{2}\sigma}. B inv = C γ ( h s ) d /2 σ m e C γ 2 σ . Here the derivatives of the cutoff enter only the prefactor C γ C_{\gamma} C γ γ \gamma γ ∣ g T ( X ) ∣ 2 ≤ C B i n v 2 ( J h ∗ A T 0 T ) ( X ) |g_{T}(X)|^{2}\le CB_{\mathrm{inv}}^{2}(J_{h}*A_{T}^{0}{}_{T})(X) ∣ g T ( X ) ∣ 2 ≤ C B inv 2 ( J h ∗ A T 0 T ) ( X ) J h ( X ′ − Y ) J_{h}(X'-Y) J h ( X ′ − Y ) J h ( X − Y ) J_{h}(X-Y) J h ( X − Y ) ∣ X ′ − X ∣ ≤ h |X'-X|\le h ∣ X ′ − X ∣ ≤ h
m X = C γ ( h s ) − d σ − m e − C 2 γ 2 σ ∣ g T ( X ) ∣ 2 . (56) m_{X}=C_{\gamma}(hs)^{-d}\sigma^{-m}e^{-C_{2}\gamma^{2}\sigma}|g_{T}(X)|^{2}. \tag*{(56)} m X = C γ ( h s ) − d σ − m e − C 2 γ 2 σ ∣ g T ( X ) ∣ 2 . (56) For the interior lower bound on − u -u − u above u u u l l l v b = ϵ m X 1 / d ( l 2 − h 2 ) v_b=\epsilon m_X^{1/d}(l^2-h^2) v b = ϵ m X 1/ d ( l 2 − h 2 ) 6 ϵ m X 1 / d I 6\epsilon m_X^{1/d}I 6 ϵ m X 1/ d I m X > 0 m_X>0 m X > 0 B f i b B_{\mathrm{fib}} B fib
C ( ( ϵ m X 1 / d ) n + λ ( ϵ m X 1 / d ) d ) ≤ λ m X . C\left((\epsilon m_X^{1/d})^n+\lambda(\epsilon m_X^{1/d})^d\right)\leq\lambda m_X. C ( ( ϵ m X 1/ d ) n + λ ( ϵ m X 1/ d ) d ) ≤ λ m X . with a fixed small ϵ \epsilon ϵ (38) and (56) give m X 1 / d ≤ μ 1 / d ≤ λ m_X^{1/d}\leq\mu^{1/d}\leq\lambda m X 1/ d ≤ μ 1/ d ≤ λ u ≤ 0 = v b u\leq0=v_b u ≤ 0 = v b B f i b + H f i b ( u ) B_{\mathrm{fib}}+H_{\mathrm{fib}}(u) B fib + H fib ( u ) λ μ ≥ λ m X \lambda\mu\geq\lambda m_X λ μ ≥ λ m X Theorem A.1 gives u ≤ v b u\leq v_b u ≤ v b ϵ h 2 m X 1 / d \epsilon h^2m_X^{1/d} ϵ h 2 m X 1/ d μ ≥ m X \mu\geq m_X μ ≥ m X ( − u ) μ ≥ ϵ h 2 m X 1 + 1 / d (-u)\mu\geq\epsilon h^2m_X^{1+1/d} ( − u ) μ ≥ ϵ h 2 m X 1 + 1/ d m X = 0 m_X=0 m X = 0 1 + 1 / d = q 1+1/d=q 1 + 1/ d = q m X m_X m X ∣ g T ( X ) ∣ 2 \lvert g_T(X)\rvert^2 ∣ g T ( X ) ∣ 2 2 q = p c 2q=p_c 2 q = p c
The imaginary volume of the center region is comparable to a 2 s 2 d γ 2 d a^2s^{2d}\gamma^{2d} a 2 s 2 d γ 2 d X X X F \mathcal{F} F h 2 s 2 d ( h s ) − d q = σ n − 2 h^{2s^{2d}}(hs)^{-dq}=\sigma^{n-2} h 2 s 2 d ( h s ) − d q = σ n − 2 m m m γ 2 d \gamma^{2d} γ 2 d
F ( T , τ ) ≥ c γ a 2 σ − m e − C 3 γ 2 σ ∥ g T ∥ p c p c . (57) \mathcal{F}(T,\tau)\geq c_\gamma a^2\sigma^{-m}e^{-C_3\gamma^2\sigma}\lVert g_T\rVert_{p_c}^{p_c}. \tag*{(57)} F ( T , τ ) ≥ c γ a 2 σ − m e − C 3 γ 2 σ ∥ g T ∥ p c p c . (57) Choose γ \gamma γ C 3 γ 2 < c 0 / 2 C_3\gamma^2<c_0/2 C 3 γ 2 < c 0 /2 K K K h < 1 / ( 4 K ) h<1/(4K) h < 1/ ( 4 K ) (38) . Dividing the flux bound (55) by the interior lower bound (57) gives
Δ 2 F F ≤ C γ σ − 2 σ 2 m e − ( c 0 − C 3 γ 2 ) σ . \frac{\Delta_2\mathcal{F}}{\mathcal{F}}\leq C_\gamma\sigma^{-2}\sigma^{2m}e^{-(c_0-C_3\gamma^2)\sigma}. F Δ 2 F ≤ C γ σ − 2 σ 2 m e − ( c 0 − C 3 γ 2 ) σ . The exponential absorbs the fixed power of σ \sigma σ (48) . On the two circles in the τ \tau τ 4 a / h = 4 K 4a/h=4K 4 a / h = 4 K 2 a / h = 2 K 2a/h=2K 2 a / h = 2 K (54) at the first circle gives (46) for the initial data; the lower bound (57) at the second gives (47) for the wave at time T T T
Propagation across a thin cylinder
The energy in Proposition 5.1 controls the wave on the circle ∣ τ ∣ = 2 a \lvert\tau\rvert=2a ∣ τ ∣ = 2 a 1 ≤ T ≤ 2 1\leq T\leq2 1 ≤ T ≤ 2 ∣ τ ∣ = 4 a \lvert\tau\rvert=4a ∣ τ ∣ = 4 a T = 0 T=0 T = 0 ∣ τ ∣ = 1 \lvert\tau\rvert=1 ∣ τ ∣ = 1 [57 Theorem 3.18] . We use the maximum principle to retain this comparison in the presence of a potential.
Lemma 6.1. Let 0 < b < 1 / 4 0<b<1/4 0 < b < 1/4 D = log ( 3 / b ) D=\log(3/b) D = log ( 3/ b ) V : [ b , 3 ] → [ 0 , ∞ ) V:[b,3]\to[0,\infty) V : [ b , 3 ] → [ 0 , ∞ )
∫ b 3 s log ( s / b ) V ( s ) d s ≤ M , sup 1 / 2 ≤ s ≤ 3 / 2 V ( s ) ≤ M . (58) \int_b^3 s\log(s/b)V(s)\,ds\leq M,\qquad\sup_{1/2\leq s\leq3/2}V(s)\leq M. \tag*{(58)} ∫ b 3 s log ( s / b ) V ( s ) d s ≤ M , 1/2 ≤ s ≤ 3/2 sup V ( s ) ≤ M . (58) Let E ≥ 0 E\geq0 E ≥ 0 R T × { τ ∈ R 2 : b ≤ ∣ τ ∣ ≤ 3 } \mathbb{R}_T\times\{\tau\in\mathbb{R}^2:b\leq\lvert\tau\rvert\leq3\} R T × { τ ∈ R 2 : b ≤ ∣ τ ∣ ≤ 3 } Δ T , τ E ≤ V ( ∣ τ ∣ ) E \Delta_{T,\tau}E\leq V(\lvert\tau\rvert)E Δ T , τ E ≤ V (∣ τ ∣) E
H ( T , s ) = 1 2 π ∫ 0 2 π E ( T , s ( cos θ , sin θ ) ) d θ H(T,s)=\frac{1}{2\pi}\int_0^{2\pi}E(T,s(\cos\theta,\sin\theta))\,d\theta H ( T , s ) = 2 π 1 ∫ 0 2 π E ( T , s ( cos θ , sin θ )) d θ satisfies, for b ≤ r ≤ 1 b\leq r\leq1 b ≤ r ≤ 1
H ( 0 , r ) ≥ c ( M ) log ( r / b ) D 2 ∫ 1 2 H ( T , b ) d T . (59) H(0,r)\geq c(M)\frac{\log(r/b)}{D^2}\int_1^2 H(T,b)\,dT. \tag*{(59)} H ( 0 , r ) ≥ c ( M ) D 2 log ( r / b ) ∫ 1 2 H ( T , b ) d T . (59) In particular the factor at r = 2 b r=2b r = 2 b D − 2 D^{-2} D − 2
Figure 2. The two comparisons, with radii s = ∣ τ ∣ s=|\tau| s = ∣ τ ∣ 1 ≤ T ≤ 2 1\le T\le2 1 ≤ T ≤ 2 [ log ( 3 / b ) ] − 2 [\log(3/b)]^{-2} [ log ( 3/ b ) ] − 2 b = 2 K h b=2K h b = 2 K h H H H
Proof. Averaging over rotations preserves the differential inequality, so H ( T , ∣ τ ∣ ) H(T,|\tau|) H ( T , ∣ τ ∣ ) ( − Δ T , τ + V ) H ≥ 0 (-\Delta_{T,\tau}+V)H\ge0 ( − Δ T , τ + V ) H ≥ 0 S = 1 S=1 S = 1 3 3 3 κ > 0 \kappa>0 κ > 0
u ′ ′ + s − 1 u ′ = ( V + κ 2 ) u , b < s < S , u^{\prime\prime}+s^{-1}u^{\prime}=(V+\kappa^2)u,\qquad b<s<S, u ′′ + s − 1 u ′ = ( V + κ 2 ) u , b < s < S , with endpoint values 0 , 1 0,1 0 , 1 u u u log s \log s log s x = log ( s / b ) x=\log(s/b) x = log ( s / b ) u x x = P ( x ) u u_{xx}=P(x)u u xx = P ( x ) u [ 0 , A ] [0,A] [ 0 , A ] A = log ( S / b ) A=\log(S/b) A = log ( S / b ) P ( x ) = s 2 ( V ( s ) + κ 2 ) P(x)=s^2(V(s)+\kappa^2) P ( x ) = s 2 ( V ( s ) + κ 2 ) ψ \psi ψ ψ ( 0 ) = 0 \psi(0)=0 ψ ( 0 ) = 0 ψ ′ ( 0 ) = 1 \psi'(0)=1 ψ ′ ( 0 ) = 1 ψ ( x ) ≥ x \psi(x)\ge x ψ ( x ) ≥ x
ψ ( x ) x = 1 + ∫ 0 x z ( x − z ) x P ( z ) ψ ( z ) z d z . \frac{\psi(x)}{x}=1+\int_0^x\frac{z(x-z)}{x}P(z)\frac{\psi(z)}{z}\,dz. x ψ ( x ) = 1 + ∫ 0 x x z ( x − z ) P ( z ) z ψ ( z ) d z . Since z ( x − z ) / x ≤ min ( z , A − z ) z(x-z)/x\le\min(z,A-z) z ( x − z ) / x ≤ min ( z , A − z ) ψ ( A ) / A ≤ exp ( ∫ 0 A min ( z , A − z ) P ( z ) d z ) \psi(A)/A\le\exp\left(\int_0^A\min(z,A-z)P(z)\,dz\right) ψ ( A ) / A ≤ exp ( ∫ 0 A min ( z , A − z ) P ( z ) d z ) V V V M M M κ 2 \kappa^2 κ 2 κ 2 S 2 ∫ 0 ∞ t e − 2 t d t \kappa^2S^2\int_0^\infty te^{-2t}\,dt κ 2 S 2 ∫ 0 ∞ t e − 2 t d t t = A − z t=A-z t = A − z ψ ( x ) / ψ ( A ) ≥ c x / A \psi(x)/\psi(A)\ge cx/A ψ ( x ) / ψ ( A ) ≥ c x / A x ↦ A − x x\mapsto A-x x ↦ A − x
For the outward passage integrate against sin ( π T / 3 ) \sin(\pi T/3) sin ( π T /3 ) [ 0 , 3 ] [0,3] [ 0 , 3 ] m ( s ) = ∫ 0 3 sin ( π T / 3 ) H ( T , s ) d T m(s)=\int_0^3\sin(\pi T/3)H(T,s)\,dT m ( s ) = ∫ 0 3 sin ( π T /3 ) H ( T , s ) d T
∫ 0 3 sin ( π T / 3 ) ∂ T 2 H ( T , s ) d T = − ( π / 3 ) 2 m ( s ) + ( π / 3 ) ( H ( 0 , s ) + H ( 3 , s ) ) . \int_0^3\sin(\pi T/3)\,\partial_T^2H(T,s)\,dT=-(\pi/3)^2m(s)+(\pi/3)(H(0,s)+H(3,s)). ∫ 0 3 sin ( π T /3 ) ∂ T 2 H ( T , s ) d T = − ( π /3 ) 2 m ( s ) + ( π /3 ) ( H ( 0 , s ) + H ( 3 , s )) . It follows that m ′ ′ + s − 1 m ′ ≤ ( V + π 2 / 9 ) m m^{\prime\prime}+s^{-1}m^{\prime}\le(V+\pi^2/9)m m ′′ + s − 1 m ′ ≤ ( V + π 2 /9 ) m m ( 3 ) ≥ 0 m(3)\ge0 m ( 3 ) ≥ 0 S = 3 S=3 S = 3 m ( s ) ≥ c m ( b ) log ( 3 / s ) / D m(s)\ge cm(b)\log(3/s)/D m ( s ) ≥ c m ( b ) log ( 3/ s ) / D
∫ 0 3 ∫ 3 / 4 ≤ ∣ τ ∣ ≤ 5 / 4 H ( T , ∣ τ ∣ ) d τ d T ≥ c m ( b ) / D . (60) \int_0^3\int_{3/4\le|\tau|\le5/4}H(T,|\tau|)\,d\tau\,dT\ge cm(b)/D. \tag*{(60)} ∫ 0 3 ∫ 3/4 ≤ ∣ τ ∣ ≤ 5/4 H ( T , ∣ τ ∣ ) d τ d T ≥ c m ( b ) / D . (60) On the fixed region − 2 < T < 4 -2<T<4 − 2 < T < 4 1 / 2 < ∣ τ ∣ < 3 / 2 1/2<|\tau|<3/2 1/2 < ∣ τ ∣ < 3/2 Δ H ≤ M H \Delta H\le MH Δ H ≤ M H H ( T , 1 ) ≥ c ( M ) m ( b ) / D H(T,1)\ge c(M)m(b)/D H ( T , 1 ) ≥ c ( M ) m ( b ) / D ∣ T ∣ ≤ 1 |T|\le1 ∣ T ∣ ≤ 1 v v v H ( T , ∣ τ ∣ ) cos ( M + 1 v ) H(T,|\tau|)\cos(\sqrt{M+1}\,v) H ( T , ∣ τ ∣ ) cos ( M + 1 v ) v = 0 v=0 v = 0 2 ρ 2\rho 2 ρ ( x , 0 ) (x,0) ( x , 0 ) ∣ v ∣ < ρ |v|<\rho ∣ v ∣ < ρ ρ \rho ρ x = ( T , τ ) x=(T,\tau) x = ( T , τ ) H ( x ) ≥ c ρ − 3 ∫ B ρ ( x ) H H(x)\ge c\rho^{-3}\int_{B_\rho(x)}H H ( x ) ≥ c ρ − 3 ∫ B ρ ( x ) H ρ \rho ρ [ − 1 , 3 ] × { 3 / 4 ≤ ∣ τ ∣ ≤ 5 / 4 } [-1,3]\times\{3/4\le|\tau|\le5/4\} [ − 1 , 3 ] × { 3/4 ≤ ∣ τ ∣ ≤ 5/4 } H ( x ) = H ( T , ∣ τ ∣ ) H(x)=H(T,|\tau|) H ( x ) = H ( T , ∣ τ ∣ ) ρ / 4 \rho/4 ρ /4 B , B ′ B,B' B , B ′ B ′ ⊂ B ρ ( x ) B'\subset B_\rho(x) B ′ ⊂ B ρ ( x ) x ∈ B x\in B x ∈ B inf B H ≥ c ρ − 3 ∫ B ′ H \inf_B H\ge c\rho^{-3}\int_{B'}H inf B H ≥ c ρ − 3 ∫ B ′ H (60) .
For the return passage take the radial solution with S = 1 S=1 S = 1 κ = π / 2 \kappa=\pi/2 κ = π /2 u ( b ) = 0 u(b)=0 u ( b ) = 0 u ( 1 ) = 1 u(1)=1 u ( 1 ) = 1 ∣ T ∣ < 1 |T|<1 ∣ T ∣ < 1 b < ∣ τ ∣ < 1 b<|\tau|<1 b < ∣ τ ∣ < 1 c ( M ) m ( b ) D − 1 cos ( π T / 2 ) u ( ∣ τ ∣ ) c(M)m(b)D^{-1}\cos(\pi T/2)u(|\tau|) c ( M ) m ( b ) D − 1 cos ( π T /2 ) u ( ∣ τ ∣ ) ( − Δ T , τ + V ) w = 0 (-\Delta_{T,\tau}+V)w=0 ( − Δ T , τ + V ) w = 0 ∣ T ∣ = 1 |T|=1 ∣ T ∣ = 1 ∣ τ ∣ = b |\tau|=b ∣ τ ∣ = b H H H ∣ τ ∣ = 1 |\tau|=1 ∣ τ ∣ = 1 u ( r ) u(r) u ( r )
H ( 0 , r ) ≥ c ( M ) m ( b ) D log ( r / b ) log ( 1 / b ) ( b ≤ r ≤ 1 ) . H(0,r)\ge\frac{c(M)m(b)}{D}\frac{\log(r/b)}{\log(1/b)}\qquad(b\le r\le1). H ( 0 , r ) ≥ D c ( M ) m ( b ) log ( 1/ b ) log ( r / b ) ( b ≤ r ≤ 1 ) . This proves (59) , since sin ( π T / 3 ) \sin(\pi T/3) sin ( π T /3 ) [ 1 , 2 ] [1,2] [ 1 , 2 ]
Remark 6.2. The dependence on r r r b b b V = 0 V=0 V = 0 ( 1 − cos T ) p ( s ) (1-\cos T)p(s) ( 1 − cos T ) p ( s ) p ( s ) = log ( 3 / s ) / D p(s)=\log(3/s)/D p ( s ) = log ( 3/ s ) / D 1 , 0 1,0 1 , 0 g ( s ) = s 2 ( 1 + log ( 3 / s ) ) / ( 4 D ) g(s)=s^2(1+\log(3/s))/(4D) g ( s ) = s 2 ( 1 + log ( 3/ s )) / ( 4 D ) L ( s ) = p ( s ) g ( b ) + ( 1 − p ( s ) ) g ( 3 ) L(s)=p(s)g(b)+(1-p(s))g(3) L ( s ) = p ( s ) g ( b ) + ( 1 − p ( s )) g ( 3 ) g g g Δ τ g = p ≥ 0 \Delta_\tau g=p\ge0 Δ τ g = p ≥ 0 E ( T , s ) = ( 1 − cos T ) p ( s ) + L ( s ) − g ( s ) E(T,s)=(1-\cos T)p(s)+L(s)-g(s) E ( T , s ) = ( 1 − cos T ) p ( s ) + L ( s ) − g ( s ) Δ T , τ E = ( cos T − 1 ) p ≤ 0 \Delta_{T,\tau}E=(\cos T-1)p\le0 Δ T , τ E = ( cos T − 1 ) p ≤ 0 s = b s=b s = b 1 − cos T 1-\cos T 1 − cos T g g g
E ( 0 , r ) ≤ ( 1 − p ( r ) ) g ( 3 ) = 9 4 log ( r / b ) D 2 . E(0,r)\le(1-p(r))g(3)=\frac{9}{4}\frac{\log(r/b)}{D^2}. E ( 0 , r ) ≤ ( 1 − p ( r )) g ( 3 ) = 4 9 D 2 log ( r / b ) . A fixed positive r r r r = 2 b r=2b r = 2 b
Proposition 6.3. On the spatial torus T n = R n / ( 100 Z ) n \mathbb{T}^n=\mathbb{R}^n/(100\mathbb{Z})^n T n = R n / ( 100 Z ) n f f f c 0 / h ≤ ∣ ξ ∣ ≤ C 0 / h c_0/h\le|\xi|\le C_0/h c 0 / h ≤ ∣ ξ ∣ ≤ C 0 / h 0 < h ≤ 1 0<h\le1 0 < h ≤ 1
( ∫ 1 2 ∥ U ( T ) f ∥ L p c ( T n ) p c d T ) 1 / p c ≤ C [ log ( 2 / h ) ] 2 / p c ∥ f ∥ L p c ( T n ) . (61) \left(\int_1^2\|U(T)f\|_{L^{p_c}(\mathbb{T}^n)}^{p_c}\,\mathrm{d}T\right)^{1/p_c}\le C[\log(2/h)]^{2/p_c}\|f\|_{L^{p_c}(\mathbb{T}^n)}. \tag*{(61)} ( ∫ 1 2 ∥ U ( T ) f ∥ L p c ( T n ) p c d T ) 1/ p c ≤ C [ log ( 2/ h ) ] 2/ p c ∥ f ∥ L p c ( T n ) . (61) The constant is uniform in h h h
Proof. For f ≠ 0 f\ne0 f = 0 ∥ f ∥ L 2 ( T n ) = 1 \|f\|_{L^2(\mathbb{T}^n)}=1 ∥ f ∥ L 2 ( T n ) = 1 K K K a = K h a=Kh a = K h b = 2 a b=2a b = 2 a h h h b < 1 / 4 b<1/4 b < 1/4 V ( s ) = C s − 2 e − c s / h V(s)=Cs^{-2}e^{-cs/h} V ( s ) = C s − 2 e − cs / h (58) follows from
∫ b 3 s log ( s / b ) V ( s ) d s ≤ C ∫ 1 ∞ log t t e − 2 c K t d t , \int_b^3s\log(s/b)V(s)\,\mathrm{d}s\le C\int_1^\infty\frac{\log t}{t}e^{-2cKt}\,\mathrm{d}t, ∫ b 3 s log ( s / b ) V ( s ) d s ≤ C ∫ 1 ∞ t log t e − 2 cK t d t , and V V V 1 / 2 ≤ s ≤ 3 / 2 1/2\le s\le3/2 1/2 ≤ s ≤ 3/2
H ( 0 , 2 b ) ≤ C K a 2 ∥ f ∥ L p c p c , H ( T , b ) ≥ c K a 2 ∥ U ( T ) f ∥ L p c p c . H(0,2b)\le C_Ka^2\|f\|_{L^{p_c}}^{p_c},\qquad H(T,b)\ge c_Ka^2\|U(T)f\|_{L^{p_c}}^{p_c}. H ( 0 , 2 b ) ≤ C K a 2 ∥ f ∥ L p c p c , H ( T , b ) ≥ c K a 2 ∥ U ( T ) f ∥ L p c p c . Take r = 2 b r=2b r = 2 b (59) , substitute both bounds, cancel a 2 a^2 a 2 p c p_c p c D = log ( 3 / ( 2 K h ) ) ≲ log ( 2 / h ) D=\log(3/(2Kh))\lesssim\log(2/h) D = log ( 3/ ( 2 K h )) ≲ log ( 2/ h ) K K K (61) . Homogeneity removes the normalization after this comparison. For the remaining h h h
Passage to Euclidean space and Sobolev estimates
The aperture in Proposition 6.3 is fixed. A finite partition of the sphere therefore gives the critical estimate on the whole annulus without increasing its logarithmic power. We first remove the spatial period and then account for the cost of summing annuli.
Removing the period
Proof of Theorem 1.2 . Write h = N − 1 h=N^{-1} h = N − 1 P h = φ ( h D ) P_h=\varphi(hD) P h = φ ( h D ) φ \varphi φ φ = ∑ ν φ ν \varphi=\sum_\nu\varphi_\nu φ = ∑ ν φ ν φ ν ( h D ) \varphi_\nu(hD) φ ν ( h D ) h − n φ ˇ ν ( ⋅ / h ) h^{-n}\check{\varphi}_\nu(\cdot/h) h − n φ ˇ ν ( ⋅ / h ) L 1 L^1 L 1 φ ν ( h D ) f \varphi_\nu(hD)f φ ν ( h D ) f ν \nu ν
∥ U ( T ) P h f ∥ L p c ( I × T n ) ≲ [ log ( 2 / h ) ] 2 / p c ∥ f ∥ L p c ( T n ) , 0 < h ≤ 1. (62) \lVert U(T)P_hf\rVert_{L^{p_c}(I\times\mathbb{T}^n)}\lesssim[\log(2/h)]^{2/p_c}\lVert f\rVert_{L^{p_c}(\mathbb{T}^n)},\qquad0<h\leq1. \tag*{(62)} ∥ U ( T ) P h f ∥ L p c ( I × T n ) ≲ [ log ( 2/ h ) ] 2/ p c ∥ f ∥ L p c ( T n ) , 0 < h ≤ 1. (62) The Euclidean convolution kernel of U ( T ) P h U(T)P_h U ( T ) P h
K h ( T , X ) = ( 2 π h ) − n ∫ e i ( X ⋅ η − T ∣ η ∣ ) / h φ ( η ) d η . K_h(T,X)=(2\pi h)^{-n}\int e^{i(X\cdot\eta-T|\eta|)/h}\varphi(\eta)\,d\eta. K h ( T , X ) = ( 2 π h ) − n ∫ e i ( X ⋅ η − T ∣ η ∣ ) / h φ ( η ) d η . For T ∈ I T\in I T ∈ I ∣ X ∣ ≥ 3 |X|\geq3 ∣ X ∣ ≥ 3 h h h ∣ X ∣ − 2 ≥ ∣ X ∣ / 3 |X|-2\geq|X|/3 ∣ X ∣ − 2 ≥ ∣ X ∣/3 h / ∣ X ∣ h/|X| h /∣ X ∣ M + n M+n M + n
∣ K h ( T , X ) ∣ ≤ C M h M ( 1 + ∣ X ∣ ) − M , T ∈ I , ∣ X ∣ ≥ 3. (63) |K_h(T,X)|\leq C_Mh^M(1+|X|)^{-M},\qquad T\in I,\quad|X|\geq3. \tag*{(63)} ∣ K h ( T , X ) ∣ ≤ C M h M ( 1 + ∣ X ∣ ) − M , T ∈ I , ∣ X ∣ ≥ 3. (63) Tile R n \mathbb{R}^n R n Q Q Q 10 Q 10Q 10 Q Q Q Q 1 ( 10 Q ) c f \boldsymbol{1}_{(10Q)^c}f 1 ( 10 Q ) c f C M h M ( ( 1 + ∣ ⋅ ∣ ) − M ∗ ∣ f ∣ ) ( X ) C_Mh^M((1+|\cdot|)^{-M}*|f|)(X) C M h M (( 1 + ∣ ⋅ ∣ ) − M ∗ ∣ f ∣ ) ( X ) ∣ X − Y ∣ ≥ 9 / 2 |X-Y|\geq9/2 ∣ X − Y ∣ ≥ 9/2 X ∈ Q X\in Q X ∈ Q Y ∉ 10 Q Y\notin10Q Y ∈ / 10 Q M > n M>n M > n L p c ( I × R n ) L^{p_c}(I\times\mathbb{R}^n) L p c ( I × R n ) O ( h M ) ∥ f ∥ p c O(h^M)\lVert f\rVert_{p_c} O ( h M ) ∥ f ∥ p c
Translate the center of Q Q Q f Q = f 1 10 Q f_Q=f\boldsymbol{1}_{10Q} f Q = f 1 10 Q ∥ f ∥ L p c ( 10 Q ) \lVert f\rVert_{L^{p_c}(10Q)} ∥ f ∥ L p c ( 10 Q ) K h K_h K h Q Q Q X − Y + 100 k X-Y+100k X − Y + 100 k Y ∈ 10 Q Y\in10Q Y ∈ 10 Q k ∈ Z n ∖ { 0 } k\in\mathbb{Z}^n\setminus\{0\} k ∈ Z n ∖ { 0 } ∣ X j − Y j ∣ ≤ 5.5 |X_j-Y_j|\leq5.5 ∣ X j − Y j ∣ ≤ 5.5 94.5 ∣ k ∣ 94.5|k| 94.5∣ k ∣ (63) over k k k C M h M ∥ f Q ∥ 1 C_Mh^M\lVert f_Q\rVert_1 C M h M ∥ f Q ∥ 1 10 Q 10Q 10 Q (62) now gives
∥ U ( T ) P h f Q ∥ L p c ( I × Q ) ≲ [ log ( 2 / h ) ] 2 / p c ∥ f ∥ L p c ( 10 Q ) . \lVert U(T)P_hf_Q\rVert_{L^{p_c}(I\times Q)}\lesssim[\log(2/h)]^{2/p_c}\lVert f\rVert_{L^{p_c}(10Q)}. ∥ U ( T ) P h f Q ∥ L p c ( I × Q ) ≲ [ log ( 2/ h ) ] 2/ p c ∥ f ∥ L p c ( 10 Q ) . Take the p c p_c p c Q Q Q 10 Q 10Q 10 Q log ( 2 / h ) ≍ log ( 2 + N ) \log(2/h)\asymp\log(2+N) log ( 2/ h ) ≍ log ( 2 + N ) N ≥ 2 N\geq2 N ≥ 2
Interpolation and summation
Plancherel bounds the norm of U ( T ) P h U(T)P_h U ( T ) P h L X 2 L^2_X L X 2 L T , X 2 L^2_{T,X} L T , X 2 ∥ φ ∥ ∞ \lVert\varphi\rVert_\infty ∥ φ ∥ ∞ ∣ I ∣ = 1 |I|=1 ∣ I ∣ = 1 p = ∞ p=\infty p = ∞ O ( h − d / 2 ) O(h^{-d/2}) O ( h − d /2 ) O ( h − d / 2 ) O(h^{-d/2}) O ( h − d /2 ) c h c\sqrt{h} c h η = ( h z , ρ ) \eta=(\sqrt{h}z,\rho) η = ( h z , ρ ) ρ \rho ρ T ρ / h T\rho/h T ρ / h
T h ( ρ 2 + h ∣ z ∣ 2 − ρ ) = T ∣ z ∣ 2 ρ 2 + h ∣ z ∣ 2 + ρ . \frac{T}{h}\left(\sqrt{\rho^2+h|z|^2}-\rho\right)=\frac{T|z|^2}{\sqrt{\rho^2+h|z|^2}+\rho}. h T ( ρ 2 + h ∣ z ∣ 2 − ρ ) = ρ 2 + h ∣ z ∣ 2 + ρ T ∣ z ∣ 2 . The expression on the right has uniformly bounded derivatives on the support. Integration by parts thus bounds the L 1 L^1 L 1
Set s + ( p ) = max { 0 , d / 2 − n / p } s_+(p) = \max\{0,d/2-n/p\} s + ( p ) = max { 0 , d /2 − n / p } β ( p ) = min { d ( 1 − 2 / p ) , 2 / p } \beta(p) = \min\{d(1-2/p),2/p\} β ( p ) = min { d ( 1 − 2/ p ) , 2/ p } Theorem 1.2 with the two bounds above gives
∥ U ( T ) P h f ∥ L p ( I × R n ) ≲ p h − s + ( p ) [ log ( 2 / h ) ] β ( p ) ∥ f ∥ L p ( R n ) , 2 ≤ p ≤ ∞ . (64) \|U(T)P_hf\|_{L^p(I\times\mathbb{R}^n)} \lesssim_p h^{-s_+(p)}[\log(2/h)]^{\beta(p)}\|f\|_{L^p(\mathbb{R}^n)}, \qquad2\leq p\leq\infty. \tag*{(64)} ∥ U ( T ) P h f ∥ L p ( I × R n ) ≲ p h − s + ( p ) [ log ( 2/ h ) ] β ( p ) ∥ f ∥ L p ( R n ) , 2 ≤ p ≤ ∞. (64) Indeed, above p c p_c p c h − 1 h^{-1} h − 1 ( d / 2 ) ( 1 − p c / p ) = d / 2 − n / p (d/2)(1-p_c/p)=d/2-n/p ( d /2 ) ( 1 − p c / p ) = d /2 − n / p [32 Chapters 1 and 6] .
Proof of Theorem 1.1. The gain of an arbitrary ε > 0 \varepsilon>0 ε > 0 g = J s + ( p ) + ε f g=J^{s_+(p)+\varepsilon}f g = J s + ( p ) + ε f 2 j 2^j 2 j j ≥ 1 j\geq1 j ≥ 1 (64) bounds the output by C 2 − j ε ( 1 + j ) β ( p ) ∥ g ∥ p C2^{-j\varepsilon}(1+j)^{\beta(p)}\|g\|_p C 2 − j ε ( 1 + j ) β ( p ) ∥ g ∥ p 2 j ( s + ( p ) + ε ) J − s + ( p ) − ε 2^{j(s_+(p)+\varepsilon)}J^{-s_+(p)-\varepsilon} 2 j ( s + ( p ) + ε ) J − s + ( p ) − ε e − i T ∣ ξ ∣ e^{-iT|\xi|} e − i T ∣ ξ ∣ T ∈ I T\in I T ∈ I p = 2 p=2 p = 2
The same summation can be made more precise if one wants an estimate at the exact power s + ( p ) s_+(p) s + ( p )
Corollary 7.1 (Logarithmic Sobolev estimate). Let 2 < p < ∞ 2<p<\infty 2 < p < ∞ δ p = 1 / 2 − 1 / p \delta_p=1/2-1/p δ p = 1/2 − 1/ p B ∗ ( p ) = β ( p ) + δ p B_*(p)=\beta(p)+\delta_p B ∗ ( p ) = β ( p ) + δ p J = ( 1 − Δ ) 1 / 2 J=(1-\Delta)^{1/2} J = ( 1 − Δ ) 1/2 ℓ 1 ( r ) = log ( 2 + r ) \ell_1(r)=\log(2+r) ℓ 1 ( r ) = log ( 2 + r ) ℓ 2 ( r ) = log ( 2 + ℓ 1 ( r ) ) \ell_2(r)=\log(2+\ell_1(r)) ℓ 2 ( r ) = log ( 2 + ℓ 1 ( r ))
∥ U ( T ) f ∥ L p ( I × R n ) ≲ p , B , C ∥ J s + ( p ) ℓ 1 ( J ) B ℓ 2 ( J ) C f ∥ L p ( R n ) (65) \|U(T)f\|_{L^p(I\times\mathbb{R}^n)}\lesssim_{p,B,C}\|J^{s_+(p)}\ell_1(J)^B\ell_2(J)^C f\|_{L^p(\mathbb{R}^n)} \tag*{(65)} ∥ U ( T ) f ∥ L p ( I × R n ) ≲ p , B , C ∥ J s + ( p ) ℓ 1 ( J ) B ℓ 2 ( J ) C f ∥ L p ( R n ) (65) whenever B > B ∗ ( p ) B>B_*(p) B > B ∗ ( p ) C ∈ R C\in\mathbb{R} C ∈ R B = B ∗ ( p ) B=B_*(p) B = B ∗ ( p ) C > δ p C>\delta_p C > δ p p = p c p=p_c p = p c B ∗ ( p c ) = 1 / 2 + 1 / p c B_*(p_c)=1/2+1/p_c B ∗ ( p c ) = 1/2 + 1/ p c δ p c = 1 / ( 2 n ) \delta_{p_c}=1/(2n) δ p c = 1/ ( 2 n ) p = 2 p=2 p = 2
Proof of Corollary 7.1. Let P j P_j P j ∣ ξ ∣ ≃ 2 j |\xi|\simeq2^j ∣ ξ ∣ ≃ 2 j j ≥ 1 j\geq1 j ≥ 1 P ~ j \widetilde P_j P j P j P_j P j s = s + ( p ) s=s_+(p) s = s + ( p ) g = J s ℓ 1 ( J ) B ℓ 2 ( J ) C f g=J^s\ell_1(J)^B\ell_2(J)^C f g = J s ℓ 1 ( J ) B ℓ 2 ( J ) C f j j j
A j = 2 j s ( 1 + j ) B [ log ( 2 + j ) ] C P j P ~ j ℓ 1 ( J ) − B ℓ 2 ( J ) − C . A_j=2^{js}(1+j)^B[\log(2+j)]^C P_j\widetilde P_j\ell_1(J)^{-B}\ell_2(J)^{-C}. A j = 2 j s ( 1 + j ) B [ log ( 2 + j ) ] C P j P j ℓ 1 ( J ) − B ℓ 2 ( J ) − C . If a j a_j a j A j A_j A j a j ( 2 j η ) a_j(2^j\eta) a j ( 2 j η ) A j A_j A j p ≥ 2 p\geq2 p ≥ 2
( ∑ j ∥ A j g ∥ p p ) 1 / p ≤ ∥ ( ∑ j ∣ A j g ∣ 2 ) 1 / 2 ∥ p ≲ p ∥ g ∥ p . \left(\sum_j\|A_jg\|_p^p\right)^{1/p}\leq\left\|\left(\sum_j|A_jg|^2\right)^{1/2}\right\|_p\lesssim_p\|g\|_p. ( j ∑ ∥ A j g ∥ p p ) 1/ p ≤ ( j ∑ ∣ A j g ∣ 2 ) 1/2 p ≲ p ∥ g ∥ p . The waves u j ( T ) = U ( T ) P j f u_j(T)=U(T)P_jf u j ( T ) = U ( T ) P j f T T T L p / 2 ( I × R n ) L^{p/2}(I\times\mathbb{R}^n) L p /2 ( I × R n )
∥ ∑ j u j ∥ L T , X p ≲ p ( ∑ j ∥ u j ∥ L T , X p 2 ) 1 / 2 . \left\|\sum_j u_j\right\|_{L^p_{T,X}}\lesssim_p\left(\sum_j\|u_j\|_{L^p_{T,X}}^2\right)^{1/2}. j ∑ u j L T , X p ≲ p ( j ∑ ∥ u j ∥ L T , X p 2 ) 1/2 . By (64) , the j j j C w j ∥ A j g ∥ p Cw_j\|A_jg\|_p C w j ∥ A j g ∥ p w j = ( 1 + j ) β ( p ) − B [ log ( 2 + j ) ] − C w_j=(1+j)^{\beta(p)-B}[\log(2+j)]^{-C} w j = ( 1 + j ) β ( p ) − B [ log ( 2 + j ) ] − C 1 / r = δ p 1/r=\delta_p 1/ r = δ p ℓ 2 \ell^2 ℓ 2 ∥ w ∥ ℓ r ( ∑ j ∥ A j g ∥ p p ) 1 / p \|w\|_{\ell^r}(\sum_j\|A_jg\|_p^p)^{1/p} ∥ w ∥ ℓ r ( ∑ j ∥ A j g ∥ p p ) 1/ p ℓ r \ell^r ℓ r B > B ∗ ( p ) B>B_*(p) B > B ∗ ( p ) B = B ∗ ( p ) B=B_*(p) B = B ∗ ( p ) ℓ r \ell^r ℓ r ∑ j ≥ 2 j − 1 ( log j ) − C r < ∞ \sum_{j\ge2}j^{-1}(\log j)^{-Cr}<\infty ∑ j ≥ 2 j − 1 ( log j ) − C r < ∞ C > δ p C>\delta_p C > δ p
For the remaining low-frequency cutoff χ 0 \chi_0 χ 0 e − i T ∣ ξ ∣ χ 0 ( ξ ) e^{-iT|\xi|}\chi_0(\xi) e − i T ∣ ξ ∣ χ 0 ( ξ ) T ∈ I T\in I T ∈ I ξ = 0 \xi=0 ξ = 0 (65) , first for finite frequency sums and then by density. At p = 2 p=2 p = 2 L 2 L^2 L 2 □ \square □
The condition on B B B C C C n ≥ 4 n\ge4 n ≥ 4 p > 2 + 4 / ( n − 3 ) p>2+4/(n-3) p > 2 + 4/ ( n − 3 ) (65) with B = C = 0 B=C=0 B = C = 0 [37 Theorem 1.2] .
Two obstructions to endpoint smoothing
A spherical wave that focuses at one time determines the necessary Sobolev order. At the critical exponent the wave spends comparable L p c L^{p_c} L p c [9 Section 3] , keeping the contributions from all these scales.
Proposition 8.1 (Radial focusing). Let n ≥ 2 n\ge2 n ≥ 2 d = n − 1 d=n-1 d = n − 1 I = [ 1 , 2 ] I=[1,2] I = [ 1 , 2 ] N N N f N f_N f N supp f N ^ ⊂ { 5 N / 4 < ∣ ξ ∣ < 7 N / 4 } \operatorname{supp}\widehat{f_N}\subset\{5N/4<|\xi|<7N/4\} supp f N ⊂ { 5 N /4 < ∣ ξ ∣ < 7 N /4 }
∥ U ( t ) f N ∥ L p ( R n × I ) ∥ f N ∥ L p ( R n ) ≳ n , p { 1 , 2 ≤ p < p c , ( log N ) 1 / p c , p = p c , N d / 2 − n / p , p c < p < ∞ . (66) \frac{\|U(t)f_N\|_{L^p(\mathbb{R}^n\times I)}}{\|f_N\|_{L^p(\mathbb{R}^n)}}\gtrsim_{n,p}
\begin{cases}
1, & 2\le p<p_c,\\
(\log N)^{1/p_c}, & p=p_c,\\
N^{d/2-n/p}, & p_c<p<\infty.
\end{cases} \tag*{(66)} ∥ f N ∥ L p ( R n ) ∥ U ( t ) f N ∥ L p ( R n × I ) ≳ n , p ⎩ ⎨ ⎧ 1 , ( log N ) 1/ p c , N d /2 − n / p , 2 ≤ p < p c , p = p c , p c < p < ∞. (66) Proof. Fix t 0 = 3 / 2 t_0=3/2 t 0 = 3/2 ψ ∈ C ∞ ( ( 5 / 4 , 7 / 4 ) ) \psi\in C^\infty((5/4,7/4)) ψ ∈ C ∞ (( 5/4 , 7/4 )) f N ^ ( ξ ) = e i t 0 ∣ ξ ∣ ψ ( ∣ ξ ∣ / N ) \widehat{f_N}(\xi)=e^{it_0|\xi|}\psi(|\xi|/N) f N ( ξ ) = e i t 0 ∣ ξ ∣ ψ ( ∣ ξ ∣/ N ) ∥ f N ∥ 2 ≍ N n / 2 \|f_N\|_2\asymp N^{n/2} ∥ f N ∥ 2 ≍ N n /2 f N f_N f N ξ = N σ ω \xi=N\sigma\omega ξ = N σ ω ∣ x ∣ < t 0 / 2 |x|<t_0/2 ∣ x ∣ < t 0 /2 t 0 + x ⋅ ω ≥ t 0 / 2 t_0+x\cdot\omega\ge t_0/2 t 0 + x ⋅ ω ≥ t 0 /2 σ \sigma σ O m ( N n − m ) O_m(N^{n-m}) O m ( N n − m ) m ≥ 0 m\ge0 m ≥ 0 ∣ x ∣ ≥ t 0 / 2 |x|\ge t_0/2 ∣ x ∣ ≥ t 0 /2 O ( ( 1 + N σ ∣ x ∣ ) − d / 2 ) O((1+N\sigma|x|)^{-d/2}) O (( 1 + N σ ∣ x ∣ ) − d /2 ) ∥ f N ∥ ∞ ≲ N ( n + 1 ) / 2 \|f_N\|_\infty\lesssim N^{(n+1)/2} ∥ f N ∥ ∞ ≲ N ( n + 1 ) /2
∥ f N ∥ p ≤ ∥ f N ∥ 2 2 / p ∥ f N ∥ ∞ 1 − 2 / p ≲ N ( n + 1 ) / 2 − 1 / p ( 2 ≤ p < ∞ ) . (67) \|f_N\|_p\le\|f_N\|_2^{2/p}\|f_N\|_\infty^{1-2/p}\lesssim N^{(n+1)/2-1/p}\qquad(2\le p<\infty). \tag*{(67)} ∥ f N ∥ p ≤ ∥ f N ∥ 2 2/ p ∥ f N ∥ ∞ 1 − 2/ p ≲ N ( n + 1 ) /2 − 1/ p ( 2 ≤ p < ∞ ) . (67) The wave at t = t 0 + u t=t_0+u t = t 0 + u x ⋅ ξ − u ∣ ξ ∣ x\cdot\xi-u|\xi| x ⋅ ξ − u ∣ ξ ∣ r = ∣ x ∣ r=|x| r = ∣ x ∣ F ( v ) = ∫ σ d / 2 ψ ( σ ) e i v σ d σ F(v)=\int\sigma^{d/2}\psi(\sigma)e^{iv\sigma}\,d\sigma F ( v ) = ∫ σ d /2 ψ ( σ ) e i v σ d σ N r ≥ 1 Nr\ge1 N r ≥ 1
U ( t 0 + u ) f N ( x ) = c n N ( n + 1 ) / 2 r − d / 2 [ e − i π d / 4 F ( N ( r − u ) ) + e i π d / 4 F ( − N ( r + u ) ) + O ( ( N r ) − 1 ) ] , (68) U(t_0+u)f_N(x)=c_nN^{(n+1)/2}r^{-d/2}\left[e^{-i\pi d/4}F\bigl(N(r-u)\bigr)+e^{i\pi d/4}F\bigl(-N(r+u)\bigr)+O\bigl((Nr)^{-1}\bigr)\right], \tag*{(68)} U ( t 0 + u ) f N ( x ) = c n N ( n + 1 ) /2 r − d /2 [ e − iπ d /4 F ( N ( r − u ) ) + e iπ d /4 F ( − N ( r + u ) ) + O ( ( N r ) − 1 ) ] , (68) where c n > 0 c_n>0 c n > 0 ± N σ r \pm N\sigma r ± N σ r ( 2 π ) d / 2 ( N σ r ) − d / 2 e ∓ i π d / 4 (2\pi)^{d/2}(N\sigma r)^{-d/2}e^{\mp i\pi d/4} ( 2 π ) d /2 ( N σ r ) − d /2 e ∓ iπ d /4 O ( ( N σ r ) − d / 2 − 1 ) O((N\sigma r)^{-d/2-1}) O (( N σ r ) − d /2 − 1 ) [74 Chapter VIII] . Since F ( 0 ) > 0 F(0)>0 F ( 0 ) > 0 F F F η > 0 \eta>0 η > 0 A A A
A / N ≤ u ≤ 1 / 4 , ∣ r − u ∣ ≤ η / N , A/N\le u\le1/4,\qquad|r-u|\le\eta/N, A / N ≤ u ≤ 1/4 , ∣ r − u ∣ ≤ η / N , the first term in brackets is bounded below in modulus and dominates the other term and the error. The shell has spatial volume ≍ N − 1 u d \asymp N^{-1}u^d ≍ N − 1 u d (67) gives
∥ U ( t ) f N ∥ L p ( R n × I ) p ∥ f N ∥ p p ≳ ∫ A / N 1 / 4 u d − d p / 2 d u . \frac{\|U(t)f_N\|_{L^p(\mathbb{R}^n\times I)}^p}{\|f_N\|_p^p}\gtrsim\int_{A/N}^{1/4}u^{d-dp/2}\,du. ∥ f N ∥ p p ∥ U ( t ) f N ∥ L p ( R n × I ) p ≳ ∫ A / N 1/4 u d − d p /2 d u . The exponent of u u u − 1 -1 − 1 p = p c p=p_c p = p c (66) . □ \square □
On this annulus J s = ( 1 − Δ ) s / 2 J^s=(1-\Delta)^{s/2} J s = ( 1 − Δ ) s /2 O s ( N s ) O_s(N^s) O s ( N s ) s s s N s N^s N s [ log ( 2 + J ) ] β [\log(2+J)]^\beta [ log ( 2 + J ) ] β O β ( ( log N ) β ) O_\beta((\log N)^\beta) O β (( log N ) β ) s ≥ max { 0 , d / 2 − n / p } s\ge\max\{0,d/2-n/p\} s ≥ max { 0 , d /2 − n / p } p c p_c p c β ≥ 1 / p c \beta\ge1/p_c β ≥ 1/ p c f N f_N f N Theorem B.1 with angular derivatives on the input.
For general data, a second example concentrates a fixed amount of L 2 L^2 L 2
Proposition 8.2 (Weak spacetime obstruction). For 2 < p ≤ p c 2<p\le p_c 2 < p ≤ p c N N N g N g_N g N supp g N ^ ⊂ { 5 N / 4 < ∣ ξ ∣ < 7 N / 4 } \operatorname{supp}\widehat{g_N}\subset\{5N/4<|\xi|<7N/4\} supp g N ⊂ { 5 N /4 < ∣ ξ ∣ < 7 N /4 }
∥ U ( t ) g N ∥ L p , ∞ ( R n × I ) ∥ g N ∥ L p ( R n ) ≳ n , p ( log N ) d ( 1 / 2 − 1 / p ) . (69) \frac{\|U(t)g_N\|_{L^{p,\infty}(\mathbb{R}^n\times I)}}{\|g_N\|_{L^p(\mathbb{R}^n)}}\gtrsim_{n,p}(\log N)^{d(1/2-1/p)}. \tag*{(69)} ∥ g N ∥ L p ( R n ) ∥ U ( t ) g N ∥ L p , ∞ ( R n × I ) ≳ n , p ( log N ) d ( 1/2 − 1/ p ) . (69) The exponent in (69) equals 1 / p c 1/p_c 1/ p c p c p_c p c 2 < p < p c 2<p<p_c 2 < p < p c 2 < p ≤ p c 2<p\le p_c 2 < p ≤ p c P N P_N P N Theorem 1.2 whenever φ ( η ) = 1 \varphi(\eta)=1 φ ( η ) = 1 5 / 4 ≤ ∣ η ∣ ≤ 7 / 4 5/4\le|\eta|\le7/4 5/4 ≤ ∣ η ∣ ≤ 7/4 1 / p c 1/p_c 1/ p c L 2 L^2 L 2 (69) for 2 < p < p c 2<p<p_c 2 < p < p c § 15 proves Proposition 8.2 by transporting disjoint packets into overlapping tubes.
Spherical extension and Kakeya tubes
Write d = n − 1 d=n-1 d = n − 1 p = p c = 2 n / d p=p_c=2n/d p = p c = 2 n / d L ( R ) = log ( 2 + R ) L(R)=\log(2+R) L ( R ) = log ( 2 + R ) A A A
For g g g S n − 1 S^{n-1} S n − 1 E g ( X ) = ∫ S n − 1 e i X ⋅ ω g ( ω ) d σ ( ω ) Eg(X)=\int_{S^{n-1}}e^{iX\cdot\omega}g(\omega)\,d\sigma(\omega) E g ( X ) = ∫ S n − 1 e i X ⋅ ω g ( ω ) d σ ( ω ) K δ f ( ω ) K_\delta f(\omega) K δ f ( ω ) ∣ f ∣ |f| ∣ f ∣ δ \delta δ ω \omega ω
Corollary 9.1 (Spherical extension and Kakeya). For n ≥ 2 n\ge2 n ≥ 2 R ≥ 2 R\ge2 R ≥ 2 z ∈ R n z\in\mathbb{R}^n z ∈ R n 0 < δ ≤ 1 0<\delta\le1 0 < δ ≤ 1
∥ E g ∥ L p c ( B ( z , R ) ) ≤ C n [ log ( 2 + R ) ] 2 / p c ∥ g ∥ L p c ( S n − 1 ) , \|Eg\|_{L^{p_c}(B(z,R))}\le C_n[\log(2+R)]^{2/p_c}\|g\|_{L^{p_c}(S^{n-1})}, ∥ E g ∥ L p c ( B ( z , R )) ≤ C n [ log ( 2 + R ) ] 2/ p c ∥ g ∥ L p c ( S n − 1 ) , ∥ K δ f ∥ L n ( S n − 1 ) ≤ C n [ log ( 2 / δ ) ] 4 / p c ∥ f ∥ L n ( R n ) . \|K_\delta f\|_{L^n(S^{n-1})}\le C_n[\log(2/\delta)]^{4/p_c}\|f\|_{L^n(\mathbb{R}^n)}. ∥ K δ f ∥ L n ( S n − 1 ) ≤ C n [ log ( 2/ δ ) ] 4/ p c ∥ f ∥ L n ( R n ) . Every subset of R n \mathbb{R}^n R n n n n
A wave estimate with logarithmic power A A A 2 A 2A 2 A 1 − 1 / n 1-1/n 1 − 1/ n 1 / p c 1/p_c 1/ p c
In the plane, Hörmander’s estimate gives the sharp logarithmic power 1 / 4 1/4 1/4 q = 4 + 1 / log R q=4+1/\log R q = 4 + 1/ log R [38 Corollary 1.3] and apply Hölder’s inequality on the observation ball and on the circle. Constant density forces this power by stationary phase. The planar directional Kakeya maximal bound has sharp power 1 / 2 1/2 1/2 [13 Equation (1.5)] ; see also Córdoba’s theorem for rectangles [19 ] . Davies proved full Hausdorff dimension in the plane [21 ] , and Wang and Zahl proved it in three dimensions [77 Theorem 1.1] ; see also [36 ] . A separate three-dimensional maximal estimate with δ − ε \delta^{-\varepsilon} δ − ε [63 Theorem 1.1] and [62 Theorem 1.1] , respectively. Here the implication from a wave estimate to a fixed logarithmic maximal estimate works in every dimension.
Lemma 9.2. Suppose (1.2) holds with L ( N ) A L(N)^A L ( N ) A A > 0 A>0 A > 0 χ ( ∣ ξ ∣ / N ) \chi(|\xi|/N) χ ( ∣ ξ ∣/ N ) χ \chi χ ( 1 / 2 , 2 ) (1/2,2) ( 1/2 , 2 ) R ≥ 2 R\ge2 R ≥ 2 z ∈ R n z\in\mathbb{R}^n z ∈ R n g ∈ L p ( S n − 1 ) g\in L^p(S^{n-1}) g ∈ L p ( S n − 1 ) E g ( X ) = ∫ S n − 1 e i X ⋅ ω g ( ω ) d σ ( ω ) Eg(X)=\int_{S^{n-1}}e^{iX\cdot\omega}g(\omega)\,d\sigma(\omega) E g ( X ) = ∫ S n − 1 e i X ⋅ ω g ( ω ) d σ ( ω )
∥ E g ∥ L p ( B ( z , R ) ) ≲ L ( R ) A ∥ g ∥ L p ( S n − 1 ) . (70) \|Eg\|_{L^p(B(z,R))}\lesssim L(R)^A\|g\|_{L^p(S^{n-1})}. \tag*{(70)} ∥ E g ∥ L p ( B ( z , R )) ≲ L ( R ) A ∥ g ∥ L p ( S n − 1 ) . (70) Proof. Choose a nonnegative nonzero ψ ∈ C c ∞ ( ( 1 , 2 ) ) \psi\in C^\infty_c((1,2)) ψ ∈ C c ∞ (( 1 , 2 ))
M N f = ∫ ψ ( T ) e i N T U ( T ) χ ( ∣ D ∣ / N ) f d T , ∥ M N f ∥ p ≲ L ( N ) A ∥ f ∥ p M_Nf=\int\psi(T)e^{iNT}U(T)\chi(|D|/N)f\,dT,\qquad\|M_Nf\|_p\lesssim L(N)^A\|f\|_p M N f = ∫ ψ ( T ) e i N T U ( T ) χ ( ∣ D ∣/ N ) f d T , ∥ M N f ∥ p ≲ L ( N ) A ∥ f ∥ p by Hölder in T T T 1 / 2 ≤ ∣ Z ∣ ≤ 3 1/2\le|Z|\le3 1/2 ≤ ∣ Z ∣ ≤ 3
K N ( Z ) = c n N d / 2 ∣ Z ∣ − d / 2 e i N ∣ Z ∣ ψ ( ∣ Z ∣ ) + O ( N d / 2 − 1 ) , c n ≠ 0. (71) K_N(Z)=c_nN^{d/2}|Z|^{-d/2}e^{iN|Z|}\psi(|Z|)+O(N^{d/2-1}),\qquad c_n\ne0. \tag*{(71)} K N ( Z ) = c n N d /2 ∣ Z ∣ − d /2 e i N ∣ Z ∣ ψ ( ∣ Z ∣ ) + O ( N d /2 − 1 ) , c n = 0. (71) Indeed, the radial multiplier is χ ( ρ / N ) ψ ^ ( ρ − N ) \chi(\rho/N)\widehat{\psi}(\rho-N) χ ( ρ / N ) ψ ( ρ − N ) u = ρ ∣ Z ∣ u=\rho|Z| u = ρ ∣ Z ∣
∫ S n − 1 e i u ω 1 d σ ( ω ) = u − d / 2 ( c + e i u + c − e − i u ) + O ( u − d / 2 − 1 ) , c + c − ≠ 0 , \int_{S^{n-1}}e^{iu\omega_1}\,d\sigma(\omega)=u^{-d/2}(c_+e^{iu}+c_-e^{-iu})+O(u^{-d/2-1}),\qquad c_+c_-\ne0, ∫ S n − 1 e i u ω 1 d σ ( ω ) = u − d /2 ( c + e i u + c − e − i u ) + O ( u − d /2 − 1 ) , c + c − = 0 , by stationary phase at the two poles. After radial integration the remainder is O ( N d / 2 − 1 ) O(N^{d/2-1}) O ( N d /2 − 1 ) χ ( ρ / N ) ρ d / 2 \chi(\rho/N)\rho^{d/2} χ ( ρ / N ) ρ d /2 N d / 2 N^{d/2} N d /2 O ( N d / 2 − 1 ) O(N^{d/2-1}) O ( N d /2 − 1 ) ∫ ∣ s ψ ^ ( s ) ∣ d s < ∞ \int|s\widehat{\psi}(s)|\,ds<\infty ∫ ∣ s ψ ( s ) ∣ d s < ∞ ψ ( ± ∣ Z ∣ ) \psi(\pm|Z|) ψ ( ± ∣ Z ∣ )
We linearize the distance phase at the focus, as in Hörmander’s passage from oscillatory integrals to extension [38 §3, (3.8)–(3.9)] ; compare Tao’s deduction of local restriction from Bochner–Riesz estimates [75 proof of Theorem 1.1] . Take f N ( r ω ) = e − i N r ψ ( r ) g ( ω ) f_N(r\omega)=e^{-iNr}\psi(r)g(\omega) f N ( r ω ) = e − i N r ψ ( r ) g ( ω ) ∥ f N ∥ p = C ψ ∥ g ∥ p \|f_N\|_p=C_\psi\|g\|_p ∥ f N ∥ p = C ψ ∥ g ∥ p ∣ X ∣ ≤ R |X|\le R ∣ X ∣ ≤ R N ≥ 4 R N\ge4R N ≥ 4 R
N ∣ r ω − X / N ∣ = N r − X ⋅ ω + O ( R 2 / N ) ( r ∈ supp ψ ) . N|r\omega-X/N|=Nr-X\cdot\omega+O(R^2/N)\qquad(r\in\operatorname{supp}\psi). N ∣ r ω − X / N ∣ = N r − X ⋅ ω + O ( R 2 / N ) ( r ∈ supp ψ ) . Substitute this expansion in (9.2) and integrate in r , ω r,\omega r , ω O ( R / N ) O(R/N) O ( R / N )
N − d / 2 M N f N ( X / N ) = c ′ E g ( − X ) + O ( ( 1 + R 2 ) N − 1 ∥ g ∥ 1 ) , c ′ = c n ∫ r d / 2 ψ ( r ) 2 d r ≠ 0. N^{-d/2}M_Nf_N(X/N)=c'Eg(-X)+O((1+R^2)N^{-1}\|g\|_1),\qquad c'=c_n\int r^{d/2}\psi(r)^2\,dr\ne0. N − d /2 M N f N ( X / N ) = c ′ E g ( − X ) + O (( 1 + R 2 ) N − 1 ∥ g ∥ 1 ) , c ′ = c n ∫ r d /2 ψ ( r ) 2 d r = 0. Choose N = ( 4 + R ) 3 + n / p N=(4+R)^{3+n/p} N = ( 4 + R ) 3 + n / p L p ( B ( 0 , R ) ) L^p(B(0,R)) L p ( B ( 0 , R )) C ∥ g ∥ p C\|g\|_p C ∥ g ∥ p N n / p − d / 2 = 1 N^{n/p-d/2}=1 N n / p − d /2 = 1 L ( N ) ≲ n L ( R ) L(N)\lesssim_n L(R) L ( N ) ≲ n L ( R ) z = 0 z=0 z = 0 g ( ω ) g(\omega) g ( ω ) e i z ⋅ ω g ( ω ) e^{iz\cdot\omega}g(\omega) e i z ⋅ ω g ( ω )
Global extension from the sphere and the truncated cone
Lemma 9.2 gives global estimates at strictly larger output exponents. For this step any fixed power of the logarithm suffices; the tube argument in the next subsection will retain that power. Write E E E S n − 1 S^{n-1} S n − 1 Σ = { ( ρ ω , ρ ) : 1 ≤ ρ ≤ 2 , ω ∈ S n − 1 } \Sigma=\{(\rho\omega,\rho):1\le\rho\le2,\ \omega\in S^{n-1}\} Σ = {( ρ ω , ρ ) : 1 ≤ ρ ≤ 2 , ω ∈ S n − 1 }
E Σ G ( x , t ) = ∫ Σ e i ( x ⋅ ξ + t τ ) G ( ξ , τ ) d Σ ( ξ , τ ) , E_\Sigma G(x,t)=\int_\Sigma e^{i(x\cdot\xi+t\tau)}G(\xi,\tau)\,d\Sigma(\xi,\tau), E Σ G ( x , t ) = ∫ Σ e i ( x ⋅ ξ + t τ ) G ( ξ , τ ) d Σ ( ξ , τ ) , with induced surface measure.
Corollary 9.3 (Global Fourier extension). Let 1 < a ≤ ∞ 1<a\le\infty 1 < a ≤ ∞ b < ∞ b<\infty b < ∞
b > p c , b > n + 1 n − 1 a ′ , b>p_c,\qquad b>\frac{n+1}{n-1}a', b > p c , b > n − 1 n + 1 a ′ , then
∥ E g ∥ L b ( R n ) ≲ n , a , b ∥ g ∥ L a ( S n − 1 ) , (72) \|Eg\|_{L^b(\mathbb{R}^n)} \lesssim_{n,a,b} \|g\|_{L^a(S^{n-1})}, \tag*{(72)} ∥ E g ∥ L b ( R n ) ≲ n , a , b ∥ g ∥ L a ( S n − 1 ) , (72) ∥ E Σ G ∥ L b ( R n + 1 ) ≲ n , a , b ∥ G ∥ L a ( Σ ) . (73) \|E_\Sigma G\|_{L^b(\mathbb{R}^{n+1})} \lesssim_{n,a,b} \|G\|_{L^a(\Sigma)}. \tag*{(73)} ∥ E Σ G ∥ L b ( R n + 1 ) ≲ n , a , b ∥ G ∥ L a ( Σ ) . (73) These are strong estimates, including when a = ∞ a=\infty a = ∞ b = ∞ b=\infty b = ∞ 1 ≤ a ≤ ∞ 1\leq a\leq\infty 1 ≤ a ≤ ∞
The two strict inequalities give the interior of the conjectured range for these compact surfaces [61 Conjecture 1.1(i)] . The three-dimensional sphere estimate in [64 §§9.1–9.2] gives this range for S 2 S^2 S 2 L 3 L^3 L 3 δ − ε \delta^{-\varepsilon} δ − ε
Proof. The local spherical estimate in Lemma 9.2, with A = 2 / p c A=2/p_c A = 2/ p c O ε ( R ε ) O_\varepsilon(R^\varepsilon) O ε ( R ε ) ε > 0 \varepsilon>0 ε > 0 B ( 0 , R ) B(0,R) B ( 0 , R ) L p c ′ ( R n ) L^{p_c'}(\mathbb{R}^n) L p c ′ ( R n ) L p c ′ ( S n − 1 ) L^{p_c'}(S^{n-1}) L p c ′ ( S n − 1 ) 1 < p c ′ < 2 1<p_c'<2 1 < p c ′ < 2 [75 Theorem 1.2] gives global diagonal restriction at every 1 < u < p c ′ 1<u<p_c' 1 < u < p c ′ E : L s → L s E:L^s\to L^s E : L s → L s p c < s < ∞ p_c<s<\infty p c < s < ∞
If a ≥ b a\geq b a ≥ b s = b s=b s = b a < b a<b a < b s = 1 + b / a ′ s=1+b/a' s = 1 + b / a ′ p c < s < b p_c<s<b p c < s < b p c = 1 + ( n + 1 ) / ( n − 1 ) p_c=1+(n+1)/(n-1) p c = 1 + ( n + 1 ) / ( n − 1 ) E : L 1 → L ∞ E:L^1\to L^\infty E : L 1 → L ∞ E : L s → L s E:L^s\to L^s E : L s → L s θ = s / b \theta=s/b θ = s / b θ / s = 1 / b \theta/s=1/b θ / s = 1/ b 1 − θ + θ / s = 1 / a 1-\theta+\theta/s=1/a 1 − θ + θ / s = 1/ a
For the cone use Nicola’s slicing argument [61 Theorem 1.2(a)(i)] . Write G ρ ( ω ) = G ( ρ ω , ρ ) G_\rho(\omega)=G(\rho\omega,\rho) G ρ ( ω ) = G ( ρ ω , ρ ) d Σ = 2 ρ n − 1 d ρ d σ d\Sigma=\sqrt{2}\rho^{n-1}\,d\rho\,d\sigma d Σ = 2 ρ n − 1 d ρ d σ
E Σ G ( x , t ) = 2 ∫ 1 2 e i t ρ ρ n − 1 E ( G ρ ) ( ρ x ) d ρ . E_\Sigma G(x,t)=\sqrt{2}\int_1^2 e^{it\rho}\rho^{n-1}E(G_\rho)(\rho x)\,d\rho. E Σ G ( x , t ) = 2 ∫ 1 2 e i tρ ρ n − 1 E ( G ρ ) ( ρ x ) d ρ . Hausdorff–Young in t t t b ≥ b ′ b\geq b' b ≥ b ′
∥ E Σ G ∥ L x , t b ≲ ( ∫ 1 2 [ ρ n − 1 − n / b ∥ E ( G ρ ) ∥ L x b ] b ′ d ρ ) 1 / b ′ ≲ ∥ G ∥ L a ( S n − 1 ) ∥ ∥ G ρ ∥ L ρ b ′ ( [ 1 , 2 ] ) . \|E_\Sigma G\|_{L^b_{x,t}}\lesssim\left(\int_1^2\left[\rho^{n-1-n/b}\|E(G_\rho)\|_{L^b_x}\right]^{b'}\,d\rho\right)^{1/b'}\lesssim\|G\|_{L^a(S^{n-1})}\|\,\|G_\rho\|_{L^{b'}_\rho([1,2])}. ∥ E Σ G ∥ L x , t b ≲ ( ∫ 1 2 [ ρ n − 1 − n / b ∥ E ( G ρ ) ∥ L x b ] b ′ d ρ ) 1/ b ′ ≲ ∥ G ∥ L a ( S n − 1 ) ∥ ∥ G ρ ∥ L ρ b ′ ([ 1 , 2 ]) . Hölder’s inequality gives (73) because b > a ′ b>a' b > a ′ a > b ′ a>b' a > b ′ b = ∞ b=\infty b = ∞ L 1 L^1 L 1
From caps to tubes
Let T δ ( ω , x ) T_\delta(\omega,x) T δ ( ω , x ) δ \delta δ x + [ − 1 / 2 , 1 / 2 ] ω x+[-1/2,1/2]\omega x + [ − 1/2 , 1/2 ] ω δ n − 1 \delta^{n-1} δ n − 1 0 < δ ≤ 1 0<\delta\leq1 0 < δ ≤ 1 K δ f ( ω ) K_\delta f(\omega) K δ f ( ω ) x x x ∣ T δ ( ω , x ) ∣ − 1 ∫ T δ ( ω , x ) ∣ f ∣ |T_\delta(\omega,x)|^{-1}\int_{T_\delta(\omega,x)}|f| ∣ T δ ( ω , x ) ∣ − 1 ∫ T δ ( ω , x ) ∣ f ∣ f ∈ L n / ( n − 1 ) f\in L^{n/(n-1)} f ∈ L n / ( n − 1 ) L n / ( n − 1 ) L^{n/(n-1)} L n / ( n − 1 )
A cap of radius δ \delta δ δ − 2 \delta^{-2} δ − 2 δ − 1 \delta^{-1} δ − 1 [7 ] and Bourgain [13 ] puts independent signs on disjoint caps. The expected squared modulus of the sum is then the sum of the squared moduli. This accounts for the factor of two in the logarithmic exponent.
Lemma 9.4. Assume (70) and set q = p / 2 = n / ( n − 1 ) q=p/2=n/(n-1) q = p /2 = n / ( n − 1 ) 0 < δ ≤ 1 / 2 0<\delta\leq1/2 0 < δ ≤ 1/2 ω j \omega_j ω j δ \delta δ S n − 1 S^{n-1} S n − 1 x j ∈ R n x_j\in\mathbb{R}^n x j ∈ R n T j = T 2 δ ( ω j , x j ) T_j=T_{2\delta}(\omega_j,x_j) T j = T 2 δ ( ω j , x j ) a j ≥ 0 a_j\geq0 a j ≥ 0
∥ ∑ j a j 1 T j ∥ L q ( R n ) ≲ L ( 1 / δ ) 2 A ( δ d ∑ j a j q ) 1 / q . (74) \left\|\sum_j a_j\mathbf{1}_{T_j}\right\|_{L^q(\mathbb{R}^n)}
\lesssim L(1/\delta)^{2A}\left(\delta^d\sum_j a_j^q\right)^{1/q}. \tag*{(74)} j ∑ a j 1 T j L q ( R n ) ≲ L ( 1/ δ ) 2 A ( δ d j ∑ a j q ) 1/ q . (74) Proof. Take the disjoint caps C j = { ∣ ω − ω j ∣ < c δ } C_j=\{|\omega-\omega_j|<c\delta\} C j = { ∣ ω − ω j ∣ < cδ } c > 0 c>0 c > 0 h j ( ω ) = e − i δ − 2 x j ⋅ ω 1 C j ( ω ) h_j(\omega)=e^{-i\delta^{-2}x_j\cdot\omega}\mathbf{1}_{C_j}(\omega) h j ( ω ) = e − i δ − 2 x j ⋅ ω 1 C j ( ω ) Y ∈ T j Y\in T_j Y ∈ T j Y − x j = t ω j + v Y-x_j=t\omega_j+v Y − x j = t ω j + v ∣ t ∣ ≤ 1 / 2 |t|\le1/2 ∣ t ∣ ≤ 1/2 ∣ v ∣ < 2 δ |v|<2\delta ∣ v ∣ < 2 δ ω j ⋅ ( ω − ω j ) = − ∣ ω − ω j ∣ 2 / 2 \omega_j\cdot(\omega-\omega_j)=-|\omega-\omega_j|^2/2 ω j ⋅ ( ω − ω j ) = − ∣ ω − ω j ∣ 2 /2 E h j ( δ − 2 Y ) Eh_j(\delta^{-2}Y) E h j ( δ − 2 Y ) ω j \omega_j ω j c 2 / 4 + 2 c c^2/4+2c c 2 /4 + 2 c ∣ E h j ( δ − 2 Y ) ∣ ≳ σ ( C j ) ≍ δ d |Eh_j(\delta^{-2}Y)|\gtrsim\sigma(C_j)\asymp\delta^d ∣ E h j ( δ − 2 Y ) ∣ ≳ σ ( C j ) ≍ δ d T j T_j T j
Group the centers by the unit lattice cubes Q Q Q J Q = { j : x j ∈ Q } J_Q=\{j:x_j\in Q\} J Q = { j : x j ∈ Q } 5 Q 5Q 5 Q C n δ − 2 C_n\delta^{-2} C n δ − 2 ε j \varepsilon_j ε j g ε = ∑ j ∈ J Q ε j a j 1 / 2 h j g_\varepsilon=\sum_{j\in J_Q}\varepsilon_j a_j^{1/2}h_j g ε = ∑ j ∈ J Q ε j a j 1/2 h j ∥ g ε ∥ p p ≲ δ d ∑ j ∈ J Q a j q \|g_\varepsilon\|_p^p\lesssim\delta^d\sum_{j\in J_Q}a_j^q ∥ g ε ∥ p p ≲ δ d ∑ j ∈ J Q a j q E ∣ E g ε ∣ p ≥ ( ∑ j ∈ J Q a j ∣ E h j ∣ ) q \mathbb{E}|Eg_\varepsilon|^p\ge\left(\sum_{j\in J_Q}a_j|Eh_j|\right)^q E ∣ E g ε ∣ p ≥ ( ∑ j ∈ J Q a j ∣ E h j ∣ ) q δ − 2 ( 5 Q ) \delta^{-2}(5Q) δ − 2 ( 5 Q ) Y Y Y (70) on the enclosing ball:
δ d p − 2 n ∫ 5 Q ( ∑ j ∈ J Q a j 1 T j ) q ≲ L ( 1 / δ ) A p δ d ∑ j ∈ J Q a j q . \delta^{dp-2n}\int_{5Q}\left(\sum_{j\in J_Q}a_j\mathbf{1}_{T_j}\right)^q\lesssim L(1/\delta)^{Ap}\delta^d\sum_{j\in J_Q}a_j^q. δ d p − 2 n ∫ 5 Q j ∈ J Q ∑ a j 1 T j q ≲ L ( 1/ δ ) A p δ d j ∈ J Q ∑ a j q . The prefactor is one because d p = 2 n dp=2n d p = 2 n 5 Q 5Q 5 Q q q q (74) is bounded by a constant times the sum of these integrals. Taking the q q q A p / q = 2 A A p/q=2A A p / q = 2 A
The maximal estimate and Hausdorff dimension
Proof of Corollary 9.1. Theorem 1.2 gives A = 2 / p A=2/p A = 2/ p A A A 0 < δ ≤ 1 / 2 0<\delta\le1/2 0 < δ ≤ 1/2 δ \delta δ { ω j } \{\omega_j\} { ω j } ∣ ω − ω j ∣ ≤ δ |\omega-\omega_j|\le\delta ∣ ω − ω j ∣ ≤ δ T δ ( ω , x ) ⊂ T 2 δ ( ω j , x ) T_\delta(\omega,x)\subset T_{2\delta}(\omega_j,x) T δ ( ω , x ) ⊂ T 2 δ ( ω j , x )
∥ K δ f ∥ L n ( S n − 1 ) ≲ ( δ d ∑ j K 2 δ f ( ω j ) n ) 1 / n . \|K_\delta f\|_{L^n(S^{n-1})}\lesssim\left(\delta^d\sum_j K_{2\delta}f(\omega_j)^n\right)^{1/n}. ∥ K δ f ∥ L n ( S n − 1 ) ≲ ( δ d j ∑ K 2 δ f ( ω j ) n ) 1/ n . To bound this discrete norm, test by a j ≥ 0 a_j\ge0 a j ≥ 0 δ d ∑ j a j q = 1 \delta^d\sum_j a_j^q=1 δ d ∑ j a j q = 1
δ d ∑ j a j ∣ T j ∣ ∫ T j ∣ f ∣ ≲ ∫ ∣ f ∣ ∑ j a j 1 T j ≲ L ( 1 / δ ) 2 A ∥ f ∥ n \delta^d\sum_j\frac{a_j}{|T_j|}\int_{T_j}|f|\lesssim\int|f|\sum_j a_j\mathbf{1}_{T_j}\lesssim L(1/\delta)^{2A}\|f\|_n δ d j ∑ ∣ T j ∣ a j ∫ T j ∣ f ∣ ≲ ∫ ∣ f ∣ j ∑ a j 1 T j ≲ L ( 1/ δ ) 2 A ∥ f ∥ n by Lemma 9.4 and Hölder. Taking suprema over the centers and then the weights proves the maximal estimate. For δ ∈ [ 1 / 2 , 1 ] \delta\in[1/2,1] δ ∈ [ 1/2 , 1 ]
Suppose F F F δ \delta δ F δ F_\delta F δ
∣ F δ ∣ ≳ L ( 1 / δ ) − 2 A n = L ( 1 / δ ) − 2 ( n − 1 ) ( A = 2 / p ) . (75) |F_\delta|\gtrsim L(1/\delta)^{-2An}=L(1/\delta)^{-2(n-1)}\qquad(A=2/p). \tag*{(75)} ∣ F δ ∣ ≳ L ( 1/ δ ) − 2 A n = L ( 1/ δ ) − 2 ( n − 1 ) ( A = 2/ p ) . (75) For bounded sets this already gives full lower and upper Minkowski dimension. To obtain Hausdorff dimension we have to allow balls of different sizes in the same cover. In fact the logarithmic rate gives infinite Hausdorff measure for every gauge
h γ ( r ) = r n [ log ( 1 / r ) ] 2 A n [ log log ( 1 / r ) ] γ , γ > n − 1 ( r small ) . h_\gamma(r)=r^n[\log(1/r)]^{2An}[\log\log(1/r)]^\gamma,\qquad\gamma>n-1\quad(r\ \text{small}). h γ ( r ) = r n [ log ( 1/ r ) ] 2 A n [ log log ( 1/ r ) ] γ , γ > n − 1 ( r small ) . Since h γ ( 2 r ) ≲ h γ ( r ) h_\gamma(2r)\lesssim h_\gamma(r) h γ ( 2 r ) ≲ h γ ( r ) { B j } \{B_j\} { B j } F F F r j ≤ e − 2 m r_j\le e^{-2^m} r j ≤ e − 2 m J k = { j : 2 k ≤ log ( 1 / r j ) < 2 k + 1 } J_k=\{j:2^k\le\log(1/r_j)<2^{k+1}\} J k = { j : 2 k ≤ log ( 1/ r j ) < 2 k + 1 } [44 Claim 3.2] . Put δ k = e − 2 k + 1 \delta_k=e^{-2^{k+1}} δ k = e − 2 k + 1 G k = ⋃ j ∈ J k 2 B j G_k=\bigcup_{j\in J_k}2B_j G k = ⋃ j ∈ J k 2 B j v k = ∑ j ∈ J k r j n v_k=\sum_{j\in J_k}r_j^n v k = ∑ j ∈ J k r j n ω \omega ω k k k C K δ k 1 G k ( ω ) CK\delta_k\mathbf{1}_{G_k}(\omega) C K δ k 1 G k ( ω ) δ k \delta_k δ k 1 ≲ ∑ k ≥ m K δ k 1 G k ( ω ) 1 \lesssim\sum_{k\ge m} K_{\delta_k}\mathbf{1}_{G_k}(\omega) 1 ≲ ∑ k ≥ m K δ k 1 G k ( ω ) k k k
1 ≲ ∑ k ≥ m 2 2 A k v k 1 / n ≲ ( ∑ j h γ ( r j ) ) 1 / n ( ∑ k ≥ m k − γ / ( n − 1 ) ) ( n − 1 ) / n . 1 \lesssim\sum_{k\ge m} 2^{2Ak}v_k^{1/n} \lesssim\left(\sum_j h_\gamma(r_j)\right)^{1/n}\left(\sum_{k\ge m} k^{-\gamma/(n-1)}\right)^{(n-1)/n}. 1 ≲ k ≥ m ∑ 2 2 A k v k 1/ n ≲ ( j ∑ h γ ( r j ) ) 1/ n ( k ≥ m ∑ k − γ / ( n − 1 ) ) ( n − 1 ) / n . The last factor tends to zero. Thus the Hausdorff measure for this gauge is infinite, and h γ ( r ) = o ( r s ) h_\gamma(r)=o(r^s) h γ ( r ) = o ( r s ) s < n s<n s < n dim H F = n \dim_{\mathrm H} F=n dim H F = n L 2 L^2 L 2 1 / 2 1/2 1/2 r 2 log ( 1 / r ) [ log log ( 1 / r ) ] γ r^2\log(1/r)[\log\log(1/r)]^\gamma r 2 log ( 1/ r ) [ log log ( 1/ r ) ] γ γ > 1 \gamma>1 γ > 1 F F F λ > 0 \lambda>0 λ > 0
The radial test f δ ( X ) = ∣ X ∣ − 1 1 { 2 δ < ∣ X ∣ < 1 / 2 } f_\delta(X)=|X|^{-1}\mathbf{1}_{\{2\delta<|X|<1/2\}} f δ ( X ) = ∣ X ∣ − 1 1 { 2 δ < ∣ X ∣ < 1/2 } L n L^n L n [ log ( 1 / δ ) ] 1 / n [\log(1/\delta)]^{1/n} [ log ( 1/ δ ) ] 1/ n δ \delta δ c ∫ δ / 4 1 / 4 t − 1 d t c\int_{\delta/4}^{1/4}t^{-1}\,dt c ∫ δ /4 1/4 t − 1 d t 1 − 1 / n = 2 / p 1-1/n=2/p 1 − 1/ n = 2/ p A = 1 / p A=1/p A = 1/ p ∣ F δ ∣ ≳ L ( 1 / δ ) − ( n − 1 ) |F_\delta|\gtrsim L(1/\delta)^{-(n-1)} ∣ F δ ∣ ≳ L ( 1/ δ ) − ( n − 1 ) [24 Theorem 1.3] .
Two consequences for other families of segments
For the Nikodym maximal function N δ f ( x ) N_\delta f(x) N δ f ( x ) through x x x
∥ N δ f ∥ L n ( R n ) ≲ L ( 1 / δ ) 2 A ∥ f ∥ L n ( R n ) . (76) \|N_\delta f\|_{L^n(\mathbb{R}^n)}\lesssim L(1/\delta)^{2A}\|f\|_{L^n(\mathbb{R}^n)}. \tag*{(76)} ∥ N δ f ∥ L n ( R n ) ≲ L ( 1/ δ ) 2 A ∥ f ∥ L n ( R n ) . (76) We give the projective argument to keep track of the logarithm; compare Tao [75, Theorem 4.10] and Gao–Liu–Xi [30, §2.2]. First consider only the ρ \rho ρ { x + u ω : 1 / 2 ≤ u ≤ 1 } \{x+u\omega:1/2\le u\le1\} { x + u ω : 1/2 ≤ u ≤ 1 } D ρ f ( x ) D_\rho f(x) D ρ f ( x ) ∣ ω n ∣ ≥ c n |\omega_n|\ge c_n ∣ ω n ∣ ≥ c n x = ( x ′ , t 0 ) x=(x',t_0) x = ( x ′ , t 0 ) Q Q Q t 0 t_0 t 0 ρ \rho ρ c n / 4 ≤ ∣ s − t 0 ∣ ≤ 2 c_n/4\le|s-t_0|\le2 c n /4 ≤ ∣ s − t 0 ∣ ≤ 2 ∣ y ′ ∣ ≤ C n |y'|\le C_n ∣ y ′ ∣ ≤ C n s − t 0 s-t_0 s − t 0
( y ′ , s ) ⟼ ( Y ′ , Y n ) = ( y ′ / ( s − t 0 ) , 1 / ( s − t 0 ) ) (y',s)\longmapsto(Y',Y_n)=\left(y'/(s-t_0),1/(s-t_0)\right) ( y ′ , s ) ⟼ ( Y ′ , Y n ) = ( y ′ / ( s − t 0 ) , 1/ ( s − t 0 ) ) and its inverse are uniformly Lipschitz there. It sends the line y ′ = x ′ + ( s − t 0 ) v y'=x'+(s-t_0)v y ′ = x ′ + ( s − t 0 ) v Y ′ = v + Y n x ′ Y'=v+Y_nx' Y ′ = v + Y n x ′ ( x ′ , 1 ) / ∣ ( x ′ , 1 ) ∣ (x',1)/|(x',1)| ( x ′ , 1 ) /∣ ( x ′ , 1 ) ∣ v v v C n ρ C_n\rho C n ρ f f f 5 Q ∩ { c n / 4 ≤ ± ( s − t 0 ) ≤ 2 } 5Q\cap\{c_n/4\le\pm(s-t_0)\le2\} 5 Q ∩ { c n /4 ≤ ± ( s − t 0 ) ≤ 2 } x ′ x' x ′ L n L^n L n C ∥ f ∥ L n ( 5 Q ) C\|f\|_{L^n(5Q)} C ∥ f ∥ L n ( 5 Q ) x ′ x' x ′ t 0 t_0 t 0 ∥ D ρ f ∥ n ≲ L ( 1 / ρ ) 2 A ∥ f ∥ n \|D_\rho f\|_n\lesssim L(1/\rho)^{2A}\|f\|_n ∥ D ρ f ∥ n ≲ L ( 1/ ρ ) 2 A ∥ f ∥ n ρ \rho ρ x x x
Return to an arbitrary segment through x x x O ( δ ) O(\delta) O ( δ ) x x x s / 2 ≤ u ≤ s s/2\le u\le s s /2 ≤ u ≤ s δ ≤ s ≤ 1 \delta\le s\le1 δ ≤ s ≤ 1 O ( δ ) O(\delta) O ( δ ) s s s O ( s ) O(s) O ( s ) D δ / s D_{\delta/s} D δ / s C δ + C ∑ s s L ( s / δ ) 2 A ≲ L ( 1 / δ ) 2 A C\delta+C\sum_s sL(s/\delta)^{2A}\lesssim L(1/\delta)^{2A} C δ + C ∑ s s L ( s / δ ) 2 A ≲ L ( 1/ δ ) 2 A (76) . The transverse disk proof above now applies to any Borel set meeting a suitable unit segment through each point of a positive measure set in length at least λ > 0 \lambda>0 λ > 0 n n n [30 Proposition 2.3] .
For a different application, fix n ≥ 3 n \ge3 n ≥ 3 d × d d \times d d × d B B B y ∈ [ − 1 , 1 ] d y \in[-1,1]^d y ∈ [ − 1 , 1 ] d
{ ( w ( y ) + ( t I d + t 2 B ) y , t ) : ∣ t ∣ ≤ τ B } . \left\{ \left(w(y) + (tI_d + t^2B)y,t\right) : |t| \le\tau_B \right\}. { ( w ( y ) + ( t I d + t 2 B ) y , t ) : ∣ t ∣ ≤ τ B } . where τ B > 0 \tau_B > 0 τ B > 0 w ( y ) w(y) w ( y ) [59 Equation (1.16)] . Nadjimzadah’s lifting theorem [59 Theorem 1.3] now applies, since Corollary 9.1 gives full Kakeya dimension in n + rank ( B 2 ) n + \operatorname{rank}(B^2) n + rank ( B 2 )
n − rank B + rank ( B 2 ) = n − dim ( im B ∩ ker B ) ; n - \operatorname{rank} B + \operatorname{rank}(B^2) = n - \dim(\operatorname{im} B \cap\ker B); n − rank B + rank ( B 2 ) = n − dim ( im B ∩ ker B ) ; the last equality is rank–nullity for B B B
Bochner–Riesz means
The Bochner–Riesz mean B r α f B_r^\alpha f B r α f ( 1 − ∣ ξ ∣ 2 / r 2 ) + α (1 - |\xi|^2/r^2)_+^\alpha ( 1 − ∣ ξ ∣ 2 / r 2 ) + α B ∗ α f = sup r > 0 ∣ B r α f ∣ B_*^\alpha f = \sup_{r>0}|B_r^\alpha f| B ∗ α f = sup r > 0 ∣ B r α f ∣ δ \delta δ δ − 1 \delta^{-1} δ − 1 [72 Equations (4.5)–(4.6)] ; Beltran, Hickman and Sogge give both the fixed-radius and maximal reductions [8 Propositions 39 and 40] . In the range p ≥ 2 p \ge2 p ≥ 2 [17 ] .
Theorem 10.1 (Bochner–Riesz means). Let n ≥ 2 n \ge2 n ≥ 2 1 ≤ p ≤ ∞ 1 \le p \le\infty 1 ≤ p ≤ ∞ δ ( p ) = max { 0 , n ∣ 1 / p − 1 / 2 ∣ − 1 / 2 } \delta(p) = \max\{0,n|1/p - 1/2| - 1/2\} δ ( p ) = max { 0 , n ∣1/ p − 1/2∣ − 1/2 } α > δ ( p ) \alpha> \delta(p) α > δ ( p )
sup r > 0 ∥ B r α f ∥ L p ( R n ) ≤ C n , p , α ∥ f ∥ L p ( R n ) . (77) \sup_{r>0}\|B_r^\alpha f\|_{L^p(\mathbb{R}^n)} \le C_{n,p,\alpha}\|f\|_{L^p(\mathbb{R}^n)}. \tag*{(77)} r > 0 sup ∥ B r α f ∥ L p ( R n ) ≤ C n , p , α ∥ f ∥ L p ( R n ) . (77) In this range B r α f → f B_r^\alpha f \to f B r α f → f L p L^p L p r → ∞ r \to\infty r → ∞ p < ∞ p < \infty p < ∞ 2 ≤ p < ∞ 2 \le p < \infty 2 ≤ p < ∞ α > s + ( p ) \alpha> s_+(p) α > s + ( p )
∥ B ∗ α f ∥ L p ( R n ) ≤ C n , p , α ∥ f ∥ L p ( R n ) , (78) \|B_*^\alpha f\|_{L^p(\mathbb{R}^n)} \le C_{n,p,\alpha}\|f\|_{L^p(\mathbb{R}^n)}, \tag*{(78)} ∥ B ∗ α f ∥ L p ( R n ) ≤ C n , p , α ∥ f ∥ L p ( R n ) , (78) and B r α f → f B_r^\alpha f \to f B r α f → f r → ∞ r \to\infty r → ∞
Here and in the square function estimate of § 11 , the strict inequality on the order of the means leaves room for a small Sobolev loss. We rescale Theorem 1.1 to a unit frequency annulus and a long time interval. For 2 ≤ p < ∞ 2 \le p < \infty 2 ≤ p < ∞ σ > s + ( p ) = max { 0 , ( n − 1 ) / 2 − n / p } \sigma> s_+(p) = \max\{0,(n - 1)/2 - n/p\} σ > s + ( p ) = max { 0 , ( n − 1 ) /2 − n / p } g ^ \widehat{g} g c ≤ ∣ ξ ∣ ≤ C c \le|\xi| \le C c ≤ ∣ ξ ∣ ≤ C
∥ U ( t ) g ∥ L p ( R n × { R ≤ ∣ t ∣ ≤ 2 R } ) ≲ c , C , n , p , σ R 1 / p + σ ∥ g ∥ p , R ≥ 1. (79) \|U(t)g\|_{L^p(\mathbb{R}^n \times\{R \le|t| \le2R\})} \lesssim_{c,C,n,p,\sigma} R^{1/p+\sigma}\|g\|_p,\qquad R \ge1. \tag*{(79)} ∥ U ( t ) g ∥ L p ( R n × { R ≤ ∣ t ∣ ≤ 2 R }) ≲ c , C , n , p , σ R 1/ p + σ ∥ g ∥ p , R ≥ 1. (79) Indeed g ( R ⋅ ) g(R\cdot) g ( R ⋅ ) R R R U ( t ) [ g ( R ⋅ ) ] ( x ) = ( U ( R t ) g ) ( R x ) U(t)[g(R\cdot)](x) = (U(Rt)g)(Rx) U ( t ) [ g ( R ⋅ )] ( x ) = ( U ( R t ) g ) ( R x ) R ( n + 1 ) / p R^{(n+1)/p} R ( n + 1 ) / p R − n / p R^{-n/p} R − n / p O ( R σ ) O(R^\sigma) O ( R σ ) ∣ t ∣ ≤ 1 |t| \le1 ∣ t ∣ ≤ 1
Lemma 10.2 (Maximal estimate for a thin shell). For 0 < δ ≤ 1 / 8 0 < \delta\le1/8 0 < δ ≤ 1/8 ψ ∈ C c ∞ ( ( 1 / 2 , 2 ) ) \psi\in C^\infty_c((1/2,2)) ψ ∈ C c ∞ (( 1/2 , 2 ))
A δ , r f = ψ ( ( 1 − ∣ D ∣ / r ) / δ ) f . A_{\delta,r}f = \psi\left((1 - |D|/r)/\delta\right)f. A δ , r f = ψ ( ( 1 − ∣ D ∣/ r ) / δ ) f . For 2 ≤ p < ∞ 2 \le p < \infty 2 ≤ p < ∞ σ > s + ( p ) \sigma> s_+(p) σ > s + ( p )
∥ sup r > 0 ∣ A δ , r f ∣ ∥ p ≲ δ − σ ∥ f ∥ p . \left\|\sup_{r>0}|A_{\delta,r}f|\right\|_p \lesssim\delta^{-\sigma}\|f\|_p. r > 0 sup ∣ A δ , r f ∣ p ≲ δ − σ ∥ f ∥ p . The constant is uniform for ψ \psi ψ C c ∞ ( ( 1 / 2 , 2 ) ) C^\infty_c((1/2,2)) C c ∞ (( 1/2 , 2 ))
Proof. For 1 ≤ r ≤ 2 1 \le r \le2 1 ≤ r ≤ 2 1 / 2 ≤ ∣ ξ ∣ ≤ 2 1/2 \le|\xi| \le2 1/2 ≤ ∣ ξ ∣ ≤ 2 χ \chi χ Q k = χ ( 2 − k D ) Q_k=\chi(2^{-k}D) Q k = χ ( 2 − k D ) g = Q 0 f g=Q_0f g = Q 0 f
A δ , r f ( x ) = δ r 2 π ∫ R e i t r ψ ^ ( δ r t ) U ( t ) g ( x ) d t . A_{\delta,r}f(x)=\frac{\delta^r}{2\pi}\int_{\mathbb{R}}e^{itr}\widehat{\psi}(\delta rt)U(t)g(x)\,\mathrm{d}t. A δ , r f ( x ) = 2 π δ r ∫ R e i t r ψ ( δ r t ) U ( t ) g ( x ) d t . Taking the supremum over r ∈ [ 1 , 2 ] r\in[1,2] r ∈ [ 1 , 2 ] C M δ ∫ ( 1 + δ ∣ t ∣ ) − M ∣ U ( t ) g ( x ) ∣ d t CM\delta\int(1+\delta|t|)^{-M}|U(t)g(x)|\,\mathrm{d}t C M δ ∫ ( 1 + δ ∣ t ∣ ) − M ∣ U ( t ) g ( x ) ∣ d t R ≤ ∣ t ∣ ≤ 2 R R\le|t|\le2R R ≤ ∣ t ∣ ≤ 2 R (79) bound the L x p L_x^p L x p
C M δ ( 1 + δ R ) − M R 1 − 1 / p R 1 / p + σ ∥ g ∥ p . C_M\delta(1+\delta R)^{-M}R^{1-1/p}R^{1/p+\sigma}\|g\|_p. C M δ ( 1 + δ R ) − M R 1 − 1/ p R 1/ p + σ ∥ g ∥ p . For M > 1 + σ M>1+\sigma M > 1 + σ R ≥ 1 R\ge1 R ≥ 1 O ( δ − σ ∥ g ∥ p ) O(\delta^{-\sigma}\|g\|_p) O ( δ − σ ∥ g ∥ p ) R = δ − 1 R=\delta^{-1} R = δ − 1 ∣ t ∣ ≤ 1 |t|\le1 ∣ t ∣ ≤ 1 O ( δ ∥ g ∥ p ) O(\delta\|g\|_p) O ( δ ∥ g ∥ p )
For r = 2 k s r=2^ks r = 2 k s 1 ≤ s ≤ 2 1\le s\le2 1 ≤ s ≤ 2 g k ( y ) = ( Q k f ) ( 2 − k y ) g_k(y)=(Q_kf)(2^{-k}y) g k ( y ) = ( Q k f ) ( 2 − k y ) A δ , r f ( 2 − k y ) = A δ , s g k ( y ) A_{\delta,r}f(2^{-k}y)=A_{\delta,s}g_k(y) A δ , r f ( 2 − k y ) = A δ , s g k ( y ) 2 − k n / p 2^{-kn/p} 2 − k n / p 2 k n / p 2^{kn/p} 2 k n / p ∥ g k ∥ p \|g_k\|_p ∥ g k ∥ p L p L^p L p C δ − σ ∥ Q k f ∥ p C\delta^{-\sigma}\|Q_kf\|_p C δ − σ ∥ Q k f ∥ p ℓ p \ell^p ℓ p p ≥ 2 p\ge2 p ≥ 2
( ∑ k ∈ Z ∥ Q k f ∥ p p ) 1 / p ≤ ∥ ( ∑ k ∈ Z ∣ Q k f ∣ 2 ) 1 / 2 ∥ p ≲ ∥ f ∥ p . \left(\sum_{k\in\mathbb{Z}}\|Q_kf\|_p^p\right)^{1/p}
\le\left\|\left(\sum_{k\in\mathbb{Z}}|Q_kf|^2\right)^{1/2}\right\|_p
\lesssim\|f\|_p. ( k ∈ Z ∑ ∥ Q k f ∥ p p ) 1/ p ≤ ( k ∈ Z ∑ ∣ Q k f ∣ 2 ) 1/2 p ≲ ∥ f ∥ p . Proof of Theorem 10.1 . Take 2 ≤ p < ∞ 2\le p<\infty 2 ≤ p < ∞ s + ( p ) < σ < α s_+(p)<\sigma<\alpha s + ( p ) < σ < α ( 1 − ρ 2 ) + α (1-\rho^2)_+^\alpha ( 1 − ρ 2 ) + α η ∈ C ∞ ( ( 1 / 2 , 2 ) ) \eta\in C^\infty((1/2,2)) η ∈ C ∞ (( 1/2 , 2 )) ∑ j ∈ Z η ( 2 j v ) = 1 \sum_{j\in\mathbb{Z}}\eta(2^jv)=1 ∑ j ∈ Z η ( 2 j v ) = 1 v > 0 v>0 v > 0 v = 1 − ρ v=1-\rho v = 1 − ρ 0 ≤ ρ < 1 0\le\rho<1 0 ≤ ρ < 1 δ = 2 − j ≤ 1 / 8 \delta=2^{-j}\le1/8 δ = 2 − j ≤ 1/8
δ α ψ δ ( ( 1 − ρ ) / δ ) , ψ δ ( u ) = η ( u ) u α ( 2 − δ u ) α . \delta^\alpha\psi_\delta((1-\rho)/\delta),\qquad\psi_\delta(u)=\eta(u)u^\alpha(2-\delta u)^\alpha. δ α ψ δ (( 1 − ρ ) / δ ) , ψ δ ( u ) = η ( u ) u α ( 2 − δ u ) α . These profiles have uniform smooth seminorms. The remainder is a smooth radial multiplier of compact support: it vanishes near ρ = 1 \rho=1 ρ = 1 ( 1 − ρ 2 ) α (1-\rho^2)^\alpha ( 1 − ρ 2 ) α Lemma 10.2 bounds the other terms in (78) by the convergent sum C ∑ j ≥ 3 2 − j ( α − σ ) ∥ f ∥ p C\sum_{j\ge3}2^{-j(\alpha-\sigma)}\|f\|_p C ∑ j ≥ 3 2 − j ( α − σ ) ∥ f ∥ p
For 2 ≤ p < ∞ 2\le p<\infty 2 ≤ p < ∞ δ ( p ) = s + ( p ) \delta(p)=s_+(p) δ ( p ) = s + ( p ) (77) . Each fixed B r α B_r^\alpha B r α δ ( p ) = δ ( p ′ ) \delta(p)=\delta(p') δ ( p ) = δ ( p ′ ) (77) for 1 < p < 2 1<p<2 1 < p < 2 p = 1 , ∞ p=1,\infty p = 1 , ∞ α > ( n − 1 ) / 2 \alpha>(n-1)/2 α > ( n − 1 ) /2 ∣ x ∣ − n / 2 − α J n / 2 + α ( ∣ x ∣ ) |x|^{-n/2-\alpha}J_{n/2+\alpha}(|x|) ∣ x ∣ − n /2 − α J n /2 + α ( ∣ x ∣ ) O ( ∣ x ∣ − ( n + 1 ) / 2 − α ) O(|x|^{-(n+1)/2-\alpha}) O ( ∣ x ∣ − ( n + 1 ) /2 − α ) r n K ( r x ) r^nK(rx) r n K ( r x ) L 1 L^1 L 1
For Schwartz functions of compact Fourier support the multipliers converge to one in all derivatives on the support as r → ∞ r\to\infty r → ∞ (77) give norm convergence for p < ∞ p<\infty p < ∞ f j f_j f j ∑ j ∥ f − f j ∥ p < ∞ \sum_j\|f-f_j\|_p<\infty ∑ j ∥ f − f j ∥ p < ∞ (78) , ∑ j B r α ( f j + 1 − f j ) \sum_j B_r^\alpha(f_{j+1}-f_j) ∑ j B r α ( f j + 1 − f j ) r ↦ B r α f j ( x ) r\mapsto B_r^\alpha f_j(x) r ↦ B r α f j ( x ) r > 0 r>0 r > 0 L p L^p L p r r r (78) to f f f f j ( x ) → f ( x ) f_j(x)\to f(x) f j ( x ) → f ( x ) r r r
Stein’s square function
Stein introduced a square function to study the convergence of Bochner–Riesz means [73 §18, (18.1)] . For the Euclidean means B r α B_r^\alpha B r α ( 1 − ∣ ξ ∣ 2 / r 2 ) + α (1-|\xi|^2/r^2)_+^\alpha ( 1 − ∣ ξ ∣ 2 / r 2 ) + α
G α f ( x ) = ( ∫ 0 ∞ ∣ r ∂ r B r f ( x ) ∣ 2 d r r ) 1 / 2 . G_\alpha f(x)=\left(\int_0^\infty\left|r\partial_r B_r f(x)\right|^2\frac{dr}{r}\right)^{1/2}. G α f ( x ) = ( ∫ 0 ∞ ∣ r ∂ r B r f ( x ) ∣ 2 r d r ) 1/2 . Carbery proved the sharp bounds in the plane [16 pp. 409–416] . In dimensions n ≥ 3 n\ge3 n ≥ 3 p ≥ 2 + 4 / n p\ge2+4/n p ≥ 2 + 4/ n [50 Theorem 1.1] . Gan, Jing and Wu reached p ≥ 13 / 4 p\ge13/4 p ≥ 13/4 [26 Theorem 1.1] , and Gan, Oh and Wu further enlarged the range in dimensions n ≥ 4 n\ge4 n ≥ 4 [27 Theorem 1.1] . Here the wave estimate (79) gives the full range 2 ≤ p < ∞ 2\le p<\infty 2 ≤ p < ∞
Theorem 11.1. Let n ≥ 2 n\ge2 n ≥ 2 2 ≤ p < ∞ 2\le p<\infty 2 ≤ p < ∞
α > max { 1 2 , n ( 1 2 − 1 p ) } , \alpha>\max\left\{\frac{1}{2},n\left(\frac{1}{2}-\frac{1}{p}\right)\right\}, α > max { 2 1 , n ( 2 1 − p 1 ) } , then ∥ G α f ∥ p ≤ C n , p , α ∥ f ∥ p \|G_\alpha f\|_p\le C_{n,p,\alpha}\|f\|_p ∥ G α f ∥ p ≤ C n , p , α ∥ f ∥ p f ∈ S ( R n ) f\in\mathcal{S}(\mathbb{R}^n) f ∈ S ( R n ) G α G_\alpha G α L p ( R n ) L^p(\mathbb{R}^n) L p ( R n )
The threshold can be read off from one shell. At distance δ \delta δ ∣ ξ ∣ = s |\xi|=s ∣ ξ ∣ = s δ α − 1 \delta^{\alpha-1} δ α − 1 s s s δ 1 / 2 \delta^{1/2} δ 1/2 δ − 1 \delta^{-1} δ − 1 δ − s + ( p ) − ε \delta^{-s_+(p)-\varepsilon} δ − s + ( p ) − ε α > 1 / 2 + s + ( p ) \alpha>1/2+s_+(p) α > 1/2 + s + ( p ) 1 ≤ s ≤ 2 1\le s\le2 1 ≤ s ≤ 2 L p L^p L p
Lemma 11.2 (One interval of radii). Let 2 < p < ∞ 2<p<\infty 2 < p < ∞ ε > 0 \varepsilon>0 ε > 0 σ = s + ( p ) + ε \sigma=s_+(p)+\varepsilon σ = s + ( p ) + ε H = L 2 ( [ 1 , 2 ] , d s ) H=L^2([1,2],\,ds) H = L 2 ([ 1 , 2 ] , d s ) A ∈ C ∞ ( R × ( 0 , ∞ ) ) A\in C^\infty(\mathbb{R}\times(0,\infty)) A ∈ C ∞ ( R × ( 0 , ∞ )) φ ∈ C ∞ ( R ) \varphi\in C^\infty(\mathbb{R}) φ ∈ C ∞ ( R ) 0 < δ ≤ 1 0<\delta\le1 0 < δ ≤ 1 T δ f ( x , s ) T_\delta f(x,s) T δ f ( x , s )
A ( s , ∣ ξ ∣ ) φ ( ( s − ∣ ξ ∣ ) / δ ) A(s,|\xi|)\varphi((s-|\xi|)/\delta) A ( s , ∣ ξ ∣ ) φ (( s − ∣ ξ ∣ ) / δ ) applied to f f f
∥ T δ f ∥ L x p ( H ) ≲ δ 1 / 2 − σ ∥ f ∥ p , ∥ T δ f ∥ L x 2 ( H ) ≲ δ 1 / 2 ∥ f ∥ 2 , \|T_\delta f\|_{L_x^p(H)}\lesssim\delta^{1/2-\sigma}\|f\|_p,\qquad\|T_\delta f\|_{L_x^2(H)}\lesssim\delta^{1/2}\|f\|_2, ∥ T δ f ∥ L x p ( H ) ≲ δ 1/2 − σ ∥ f ∥ p , ∥ T δ f ∥ L x 2 ( H ) ≲ δ 1/2 ∥ f ∥ 2 , with constants independent of δ \delta δ
Proof. Temporarily omit A A A s ∈ R s\in\mathbb{R} s ∈ R V δ V_\delta V δ g g g s s s x x x
∥ V δ g ( x , ⋅ ) ∥ 2 = ( 2 π ) − 1 / 2 δ ∥ φ ^ ( δ t ) U ( t ) g ( x ) ∥ L t 2 . \|V_\delta g(x,\cdot)\|_2=(2\pi)^{-1/2}\delta\|\widehat{\varphi}(\delta t)U(t)g(x)\|_{L_t^2}. ∥ V δ g ( x , ⋅ ) ∥ 2 = ( 2 π ) − 1/2 δ ∥ φ ( δ t ) U ( t ) g ( x ) ∥ L t 2 . On R ≤ ∣ t ∣ ≤ 2 R R\le|t|\le2R R ≤ ∣ t ∣ ≤ 2 R R ≥ 1 R\ge1 R ≥ 1 t t t (79) give
∥ U ( t ) g ∥ L x p L t 2 ≲ R 1 / 2 − 1 / p R 1 / p + σ ∥ g ∥ p . \|U(t)g\|_{L_x^pL_t^2}\lesssim R^{1/2-1/p}R^{1/p+\sigma}\|g\|_p. ∥ U ( t ) g ∥ L x p L t 2 ≲ R 1/2 − 1/ p R 1/ p + σ ∥ g ∥ p . On ∣ t ∣ ≤ 1 |t|\le1 ∣ t ∣ ≤ 1 L 1 L^1 L 1 b > 1 / 2 + σ b>1/2+\sigma b > 1/2 + σ R = 1 , 2 , 4 , … R=1,2,4,\ldots R = 1 , 2 , 4 , …
∥ V δ g ∥ L x p L s 2 ( R ) ≲ δ ( 1 + ∑ R R 1 / 2 + σ ( 1 + δ R ) − b ) ∥ g ∥ p ≲ δ 1 / 2 − σ ∥ g ∥ p . \|V_\delta g\|_{L_x^pL_s^2(\mathbb{R})}\lesssim\delta\left(1+\sum_R R^{1/2+\sigma}(1+\delta R)^{-b}\right)\|g\|_p\lesssim\delta^{1/2-\sigma}\|g\|_p. ∥ V δ g ∥ L x p L s 2 ( R ) ≲ δ ( 1 + R ∑ R 1/2 + σ ( 1 + δ R ) − b ) ∥ g ∥ p ≲ δ 1/2 − σ ∥ g ∥ p . To restore A A A s s s A ^ 1 ( η , ∣ D ∣ ) \widehat{A}_1(\eta,|D|) A 1 ( η , ∣ D ∣ ) L p L^p L p O b ( ( 1 + ∣ η ∣ ) − b ) O_b((1+|\eta|)^{-b}) O b (( 1 + ∣ η ∣ ) − b ) b > 0 b>0 b > 0 e i s η V δ ( A ^ 1 ( η , ∣ D ∣ ) f ) e^{is\eta}V_\delta(\widehat{A}_1(\eta,|D|)f) e i sη V δ ( A 1 ( η , ∣ D ∣ ) f ) η \eta η d η / ( 2 π ) d\eta/(2\pi) d η / ( 2 π ) s s s O ( δ ) O(\delta) O ( δ )
To combine the intervals of radii, we use Seeger’s method for passing from bounds on individual frequency annuli to a multiplier estimate [67 Theorem 1] . Jones, Seeger and Wright stated a square function version for dilates of annular multipliers [40 p. 6737] ; Guo, Roos and Yung proved it [34 Proposition 4.2 and §7] . Here we allow a different input at each scale and assume decay of the Hilbert-valued kernel and its first derivative. On each test cube only O ( 1 + log ( 1 / δ ) ) O(1+\log(1/\delta)) O ( 1 + log ( 1/ δ )) L p L^p L p 1 / 2 − 1 / p 1/2-1/p 1/2 − 1/ p Theorem 11.1 absorbs.
Lemma 11.3 (Combining dilates). Let H H H 2 < p < ∞ 2<p<\infty 2 < p < ∞ 0 < δ ≤ 1 0<\delta\leq1 0 < δ ≤ 1 N = n + 3 N=n+3 N = n + 3 K : R n → H K:\mathbb{R}^n\to H K : R n → H
∥ K ( x ) ∥ H + ∥ ∇ K ( x ) ∥ H n ≤ C ∗ δ − N ( 1 + ∣ x ∣ ) − N . \|K(x)\|_{H}+\|\nabla K(x)\|_{H^n}\leq C_*\delta^{-N}(1+|x|)^{-N}. ∥ K ( x ) ∥ H + ∥∇ K ( x ) ∥ H n ≤ C ∗ δ − N ( 1 + ∣ x ∣ ) − N . Write S ( k ) f = 2 n k K ( 2 k ⋅ ) ∗ f S^{(k)}f=2^{nk}K(2^k\cdot)*f S ( k ) f = 2 nk K ( 2 k ⋅ ) ∗ f B q B_q B q K ∗ K* K ∗ L q L^q L q L q ( H ) L^q(H) L q ( H ) q = 2 , p q=2,p q = 2 , p f k f_k f k
∥ ( ∑ k ∥ S ( k ) f k ∥ H 2 ) 1 / 2 ∥ p ≲ n , p ( B 2 + C ∗ δ + [ 1 + log ( 1 / δ ) ] 1 / 2 − 1 / p B p ) ∥ ( ∑ k ∣ f k ∣ 2 ) 1 / 2 ∥ p . \left\|\left(\sum_k\|S^{(k)}f_k\|_H^2\right)^{1/2}\right\|_p\lesssim_{n,p}\left(B_2+C_*\delta+[1+\log(1/\delta)]^{1/2-1/p}B_p\right)\left\|\left(\sum_k|f_k|^2\right)^{1/2}\right\|_p. ( k ∑ ∥ S ( k ) f k ∥ H 2 ) 1/2 p ≲ n , p ( B 2 + C ∗ δ + [ 1 + log ( 1/ δ ) ] 1/2 − 1/ p B p ) ( k ∑ ∣ f k ∣ 2 ) 1/2 p . Proof. Put b k = ∥ S ( k ) f k ∥ H b_k=\|S^{(k)}f_k\|_H b k = ∥ S ( k ) f k ∥ H h = ( ∑ k b k 2 ) 1 / 2 h=(\sum_k b_k^2)^{1/2} h = ( ∑ k b k 2 ) 1/2 W = ( ∑ k ∣ f k ∣ 2 ) 1 / 2 W=(\sum_k|f_k|^2)^{1/2} W = ( ∑ k ∣ f k ∣ 2 ) 1/2 M M M L = ⌈ ( N + 2 ) log 2 ( 2 / δ ) ⌉ L=\lceil(N+2)\log_2(2/\delta)\rceil L = ⌈( N + 2 ) log 2 ( 2/ δ )⌉ Q Q Q 2 − j 2^{-j} 2 − j y y y k ≤ j − L k\leq j-L k ≤ j − L h h h c Q = ( ∑ k ≤ j − L b k ( y ) 2 ) 1 / 2 c_Q=(\sum_{k\leq j-L}b_k(y)^2)^{1/2} c Q = ( ∑ k ≤ j − L b k ( y ) 2 ) 1/2 x , z ∈ Q x,z\in Q x , z ∈ Q k ≤ j − L k\leq j-L k ≤ j − L
∥ S ( k ) f k ( x ) − S ( k ) f k ( y ) ∥ H ≲ C ∗ δ − N 2 k − j M W ( z ) . \|S^{(k)}f_k(x)-S^{(k)}f_k(y)\|_H\lesssim C_*\delta^{-N}2^{k-j}MW(z). ∥ S ( k ) f k ( x ) − S ( k ) f k ( y ) ∥ H ≲ C ∗ δ − N 2 k − j M W ( z ) . Indeed 2 k ∣ x − z ∣ ≲ 1 2^k|x-z|\lesssim1 2 k ∣ x − z ∣ ≲ 1 2 n k ( 1 + 2 k ∣ ⋅ ∣ ) − N 2^{nk}(1+2^k|\cdot|)^{-N} 2 nk ( 1 + 2 k ∣ ⋅ ∣ ) − N M M M C C ∗ δ M W ( z ) CC_*\delta MW(z) C C ∗ δ M W ( z )
For k ≥ j + L k\geq j+L k ≥ j + L f k f_k f k 5 Q 5Q 5 Q
1 ∣ Q ∣ ∫ Q ∑ k ≥ j + L ∥ S ( k ) ( f k 1 5 Q ) ( x ) ∥ H 2 d x ≲ B 2 2 M ( W 2 ) ( z ) , z ∈ Q , \frac{1}{|Q|}\int_Q\sum_{k\geq j+L}\|S^{(k)}(f_k\mathbf{1}_{5Q})(x)\|_H^2\,dx\lesssim B_2^2M(W^2)(z),\qquad z\in Q, ∣ Q ∣ 1 ∫ Q k ≥ j + L ∑ ∥ S ( k ) ( f k 1 5 Q ) ( x ) ∥ H 2 d x ≲ B 2 2 M ( W 2 ) ( z ) , z ∈ Q , because dilation preserves the L 2 L^2 L 2 5 Q 5Q 5 Q x , z ∈ Q x,z\in Q x , z ∈ Q
∥ S ( k ) ( f k 1 ( 5 Q ) c ) ( x ) ∥ H ≲ C ∗ δ − N 2 − ( N − n ) ( k − j ) M W ( z ) . \|S^{(k)}(f_k\mathbf{1}_{(5Q)^c})(x)\|_H\lesssim C_*\delta^{-N}2^{-(N-n)(k-j)}MW(z). ∥ S ( k ) ( f k 1 ( 5 Q ) c ) ( x ) ∥ H ≲ C ∗ δ − N 2 − ( N − n ) ( k − j ) M W ( z ) . Their square sum again has square root at most C C ∗ δ M W ( z ) CC_*\delta MW(z) C C ∗ δ M W ( z )
There remain fewer than 2 L 2L 2 L
( ∣ Q ∣ − 1 ∫ Q ∑ ∣ k − j ∣ < L b k 2 ) 1 / 2 \left(|Q|^{-1}\int_Q\sum_{|k-j|<L}b_k^2\right)^{1/2} ∣ Q ∣ − 1 ∫ Q ∣ k − j ∣ < L ∑ b k 2 1/2 by ( 2 L ) 1 / 2 − 1 / p F ( z ) (2L)^{1/2-1/p}F(z) ( 2 L ) 1/2 − 1/ p F ( z ) F = ( ∑ k [ M ( b k 2 ) ] p / 2 ) 1 / p F=(\sum_k[M(b_k^2)]^{p/2})^{1/p} F = ( ∑ k [ M ( b k 2 ) ] p /2 ) 1/ p p / 2 > 1 p/2>1 p /2 > 1 B p B_p B p
∥ F ∥ p ≲ ( ∑ k ∥ b k ∥ p p ) 1 / p ≤ B p ( ∑ k ∥ f k ∥ p p ) 1 / p ≤ B p ∥ W ∥ p . \|F\|_p\lesssim\left(\sum_k\|b_k\|_p^p\right)^{1/p}\leq B_p\left(\sum_k\|f_k\|_p^p\right)^{1/p}\leq B_p\|W\|_p. ∥ F ∥ p ≲ ( k ∑ ∥ b k ∥ p p ) 1/ p ≤ B p ( k ∑ ∥ f k ∥ p p ) 1/ p ≤ B p ∥ W ∥ p . By the triangle inequality for the square norm and Cauchy–Schwarz on Q Q Q ∣ h − c Q ∣ |h-c_Q| ∣ h − c Q ∣ C ∗ δ M W ( z ) + B 2 M ( W 2 ) ( z ) 1 / 2 + L 1 / 2 − 1 / p F ( z ) C_*\delta MW(z)+B_2M(W^2)(z)^{1/2}+L^{1/2-1/p}F(z) C ∗ δ M W ( z ) + B 2 M ( W 2 ) ( z ) 1/2 + L 1/2 − 1/ p F ( z ) c Q c_Q c Q h Q h_Q h Q [74 Chapter IV, §3.6, Corollary 1] and the maximal theorem on L p L^p L p L p / 2 L^{p/2} L p /2 ∥ h ∥ 2 2 = ∑ k ∥ S ( k ) f k ∥ L 2 ( H ) 2 ≤ B 2 2 ∑ k ∥ f k ∥ 2 2 < ∞ \|h\|_2^2=\sum_k\|S^{(k)}f_k\|_{L^2(H)}^2\leq B_2^2\sum_k\|f_k\|_2^2<\infty ∥ h ∥ 2 2 = ∑ k ∥ S ( k ) f k ∥ L 2 ( H ) 2 ≤ B 2 2 ∑ k ∥ f k ∥ 2 2 < ∞ 2 < p 2<p 2 < p
Proof of Theorem 11.1. For p = 2 p=2 p = 2 u = ∣ ξ ∣ 2 / r 2 u=|\xi|^2/r^2 u = ∣ ξ ∣ 2 / r 2
∥ G α f ∥ 2 2 = 2 α 2 ∫ 0 1 u ( 1 − u ) 2 α − 2 d u ∥ f ∥ 2 2 = α 2 α − 1 ∥ f ∥ 2 2 . \|G_\alpha f\|_2^2=2\alpha^2\int_0^1 u(1-u)^{2\alpha-2}\,du\,\|f\|_2^2=\frac{\alpha}{2\alpha-1}\|f\|_2^2. ∥ G α f ∥ 2 2 = 2 α 2 ∫ 0 1 u ( 1 − u ) 2 α − 2 d u ∥ f ∥ 2 2 = 2 α − 1 α ∥ f ∥ 2 2 . Suppose p > 2 p>2 p > 2 ε > 0 \varepsilon>0 ε > 0 σ = s + ( p ) + ε < α − 1 / 2 \sigma=s_+(p)+\varepsilon<\alpha-1/2 σ = s + ( p ) + ε < α − 1/2 1 / 2 + s + ( p ) = max { 1 / 2 , n ( 1 / 2 − 1 / p ) } 1/2+s_+(p)=\max\{1/2,n(1/2-1/p)\} 1/2 + s + ( p ) = max { 1/2 , n ( 1/2 − 1/ p )} r = 2 k s r=2^k s r = 2 k s 1 ≤ s ≤ 2 1\leq s\leq2 1 ≤ s ≤ 2 ρ = 2 − k ∣ ξ ∣ \rho=2^{-k}|\xi| ρ = 2 − k ∣ ξ ∣ d r / r = d s / s dr/r=ds/s d r / r = d s / s d s ds d s
m ( s , ρ ) = 2 α ρ 2 s − 2 α ( s + ρ ) α − 1 ( s − ρ ) α − 1 1 { ρ < s } , m(s,\rho)=2\alpha\rho^2s^{-2\alpha}(s+\rho)^{\alpha-1}(s-\rho)^{\alpha-1}\mathbf{1}_{\{\rho<s\}}, m ( s , ρ ) = 2 α ρ 2 s − 2 α ( s + ρ ) α − 1 ( s − ρ ) α − 1 1 { ρ < s } , where the last power is taken only for ρ < s \rho<s ρ < s ρ = s \rho=s ρ = s s − ρ > 0 s-\rho>0 s − ρ > 0 δ = 2 − j \delta=2^{-j} δ = 2 − j j ≥ 0 j\geq0 j ≥ 0 δ α − 1 A ( s , ρ ) φ ( ( s − ρ ) / δ ) \delta^{\alpha-1}A(s,\rho)\varphi((s-\rho)/\delta) δ α − 1 A ( s , ρ ) φ (( s − ρ ) / δ ) ( s − ρ ) α − 1 (s-\rho)^{\alpha-1} ( s − ρ ) α − 1 A A A ρ = 0 \rho=0 ρ = 0 φ ∈ C c ∞ ( ( 1 / 2 , 2 ) ) \varphi\in C^\infty_c((1/2,2)) φ ∈ C c ∞ (( 1/2 , 2 ))
For fixed δ \delta δ T δ T_\delta T δ C ∗ C_* C ∗ δ \delta δ N N N δ − N \delta^{-N} δ − N s s s T δ ( k ) T_\delta^{(k)} T δ ( k ) ∣ ξ ∣ |\xi| ∣ ξ ∣ 2 − k ∣ ξ ∣ 2^{-k}|\xi| 2 − k ∣ ξ ∣ T δ T_\delta T δ s ∈ [ 1 , 2 ] s\in[1,2] s ∈ [ 1 , 2 ] P k P_k P k P k f P_kf P k f
C δ α − 1 ( δ 1 / 2 + δ + [ 1 + log ( 1 / δ ) ] 1 / 2 − 1 / p δ 1 / 2 − σ ) ∥ f ∥ p . C\delta^{\alpha-1}\left(\delta^{1/2}+\delta+[1+\log(1/\delta)]^{1/2-1/p}\delta^{1/2-\sigma}\right)\|f\|_p. C δ α − 1 ( δ 1/2 + δ + [ 1 + log ( 1/ δ ) ] 1/2 − 1/ p δ 1/2 − σ ) ∥ f ∥ p . The sum over j j j α − 1 / 2 − σ > 0 \alpha-1/2-\sigma>0 α − 1/2 − σ > 0
The multiplier left away from the boundary is smooth, supported in ρ ≤ 2 \rho\leq2 ρ ≤ 2 ρ ≍ 2 ℓ \rho\asymp2^\ell ρ ≍ 2 ℓ ℓ ≤ 2 \ell\leq2 ℓ ≤ 2 2 2 ℓ 2^{2\ell} 2 2 ℓ δ = 1 \delta=1 δ = 1 B 2 B_2 B 2 B p B_p B p ℓ \ell ℓ 2 k + ℓ 2^{k+\ell} 2 k + ℓ ∑ ℓ ≤ 2 2 2 ℓ < ∞ \sum_{\ell\leq2}2^{2\ell}<\infty ∑ ℓ ≤ 2 2 2 ℓ < ∞
These estimates apply first to finite sets of k k k f f f r r r f ↦ ( r ∂ r B r α f ) r > 0 f\mapsto(r\partial_rB_r^\alpha f)_{r>0} f ↦ ( r ∂ r B r α f ) r > 0 L x p ( L 2 ( d r / r ) ) L^p_x(L^2(dr/r)) L x p ( L 2 ( d r / r ))
The paraboloid and Schrödinger evolution
We return to the spherical extension estimate of Lemma 9.2, with n = d + 1 n=d+1 n = d + 1 S d S^d S d [38 §3, (3.10)–(3.12)] . The Fresnel formula then writes a Schrödinger wave in d d d p c = 2 ( d + 1 ) / d p_c=2(d+1)/d p c = 2 ( d + 1 ) / d [66 ] and Lee, Rogers and Seeger [51 ] .
Write B d B_d B d R d \mathbb{R}^d R d
E g ( x , t ) = ∫ B d e i ( x ⋅ v + t ∣ v ∣ 2 ) g ( v ) d v . Eg(x,t)=\int_{B_d}e^{i(x\cdot v+t|v|^2)}g(v)\,dv. E g ( x , t ) = ∫ B d e i ( x ⋅ v + t ∣ v ∣ 2 ) g ( v ) d v . Corollary 12.1 (Paraboloid extension and Schrödinger local smoothing). Let d ≥ 1 d \ge1 d ≥ 1 p c = 2 ( d + 1 ) / d p_c = 2(d+1)/d p c = 2 ( d + 1 ) / d 1 < a ≤ ∞ 1 < a \le\infty 1 < a ≤ ∞ p c < b < ∞ p_c < b < \infty p c < b < ∞ b > ( d + 2 ) a ′ / d b > (d+2)a'/d b > ( d + 2 ) a ′ / d
∥ E g ∥ L b ( R d + 1 ) ≲ d , a , b ∥ g ∥ L a ( B d ) . (80) \|Eg\|_{L^b(\mathbb{R}^{d+1})} \lesssim_{d,a,b} \|g\|_{L^a(B_d)}. \tag*{(80)} ∥ E g ∥ L b ( R d + 1 ) ≲ d , a , b ∥ g ∥ L a ( B d ) . (80) Let I ⊂ R I \subset\mathbb{R} I ⊂ R N ≥ 2 N \ge2 N ≥ 2 f f f supp f ^ ⊂ B ( 0 , N ) ⊂ R d \operatorname{supp}\widehat{f} \subset B(0,N) \subset\mathbb{R}^d supp f ⊂ B ( 0 , N ) ⊂ R d
∥ e i t Δ f ∥ L p c ( R d × I ) ≲ d , I [ log ( 2 + N ) ] 2 / p c ∥ f ∥ p c . (81) \|e^{it\Delta}f\|_{L^{p_c}(\mathbb{R}^d \times I)} \lesssim_{d,I} [\log(2+N)]^{2/p_c}\|f\|_{p_c}. \tag*{(81)} ∥ e i t Δ f ∥ L p c ( R d × I ) ≲ d , I [ log ( 2 + N ) ] 2/ p c ∥ f ∥ p c . (81) For p c < p < ∞ p_c < p < \infty p c < p < ∞ p ≤ r ≤ ∞ p \le r \le\infty p ≤ r ≤ ∞
∥ e i t Δ f ∥ L p ( R d ; L r ( I ) ) ≲ d , p , r , I ∥ f ∥ B p , p s ( R d ) ≲ d , p , s ∥ ( 1 − Δ ) s / 2 f ∥ p , s = d ( 1 − 2 / p ) − 2 / r . (82) \|e^{it\Delta}f\|_{L^p(\mathbb{R}^d;L^r(I))} \lesssim_{d,p,r,I} \|f\|_{B^s_{p,p}(\mathbb{R}^d)} \lesssim_{d,p,s} \|(1-\Delta)^{s/2}f\|_p,\qquad s=d(1-2/p)-2/r. \tag*{(82)} ∥ e i t Δ f ∥ L p ( R d ; L r ( I )) ≲ d , p , r , I ∥ f ∥ B p , p s ( R d ) ≲ d , p , s ∥ ( 1 − Δ ) s /2 f ∥ p , s = d ( 1 − 2/ p ) − 2/ r . (82) Here B p , p s B^s_{p,p} B p , p s r = p r=p r = p d − 2 ( d + 1 ) / p d-2(d+1)/p d − 2 ( d + 1 ) / p r = ∞ r=\infty r = ∞
Proof. We first transfer Lemma 9.2 to E E E A A A A = 2 / p c A=2/p_c A = 2/ p c ( δ v , 1 − δ 2 ∣ v ∣ 2 ) (\delta v,\sqrt{1-\delta^2|v|^2}) ( δ v , 1 − δ 2 ∣ v ∣ 2 ) ( x / δ , − 2 t / δ 2 ) (x/\delta,-2t/\delta^2) ( x / δ , − 2 t / δ 2 ) q δ ( v ) = 2 ( 1 − 1 − δ 2 ∣ v ∣ 2 ) / δ 2 q_\delta(v)=2(1-\sqrt{1-\delta^2|v|^2})/\delta^2 q δ ( v ) = 2 ( 1 − 1 − δ 2 ∣ v ∣ 2 ) / δ 2 g ( v ) 1 − δ 2 ∣ v ∣ 2 g(v)\sqrt{1-\delta^2|v|^2} g ( v ) 1 − δ 2 ∣ v ∣ 2 δ d g ( v ) d v \delta^d g(v)\,dv δ d g ( v ) d v B ( 0 , R ) B(0,R) B ( 0 , R ) B ( 0 , C R / δ 2 ) B(0,CR/\delta^2) B ( 0 , C R / δ 2 ) q δ q_\delta q δ L p c → L p c L^{p_c}\to L^{p_c} L p c → L p c
C δ − d + ( d + 2 ) / p c + d / p c [ log ( 2 + R / δ 2 ) ] A . C\delta^{-d+(d+2)/p_c+d/p_c}[\log(2+R/\delta^2)]^A. C δ − d + ( d + 2 ) / p c + d / p c [ log ( 2 + R / δ 2 ) ] A . The power of δ \delta δ q δ ( v ) = ∣ v ∣ 2 + O ( δ 2 ) q_\delta(v)=|v|^2+O(\delta^2) q δ ( v ) = ∣ v ∣ 2 + O ( δ 2 ) E g Eg E g B ( 0 , R ) B(0,R) B ( 0 , R ) C R δ 2 ∥ g ∥ 1 CR\delta^2\|g\|_1 C R δ 2 ∥ g ∥ 1 δ = ( 2 R ) − ( d + 2 ) \delta=(2R)^{-(d+2)} δ = ( 2 R ) − ( d + 2 ) L p c L^{p_c} L p c O ( ∥ g ∥ p c ) O(\|g\|_{p_c}) O ( ∥ g ∥ p c ) log ( 2 + R / δ 2 ) ≲ d log ( 2 + R ) \log(2+R/\delta^2)\lesssim_d\log(2+R) log ( 2 + R / δ 2 ) ≲ d log ( 2 + R ) g g g
∥ E g ∥ L p c ( B ( z , R ) ) ≲ d , A [ log ( 2 + R ) ] A ∥ g ∥ p c ( z ∈ R d + 1 , R ≥ 1 ) . (83) \|Eg\|_{L^{p_c}(B(z,R))}\lesssim_{d,A}[\log(2+R)]^A\|g\|_{p_c}\qquad(z\in\mathbb{R}^{d+1},\ R\ge1). \tag*{(83)} ∥ E g ∥ L p c ( B ( z , R )) ≲ d , A [ log ( 2 + R ) ] A ∥ g ∥ p c ( z ∈ R d + 1 , R ≥ 1 ) . (83) To obtain (80) , first note that the loss in (83) is O ε ( R ε ) O_\varepsilon(R^\varepsilon) O ε ( R ε ) ε > 0 \varepsilon>0 ε > 0 d v dv d v B d B_d B d O ( ( 1 + ∣ ( x , t ) ∣ ) − d / 2 ) O((1+|(x,t)|)^{-d/2}) O (( 1 + ∣ ( x , t ) ∣ ) − d /2 ) [75 Theorem 1.2] in the compact hypersurface form of [46 Theorem 5.3] , with both restriction exponents equal to p c ′ ∈ ( 1 , 2 ) p'_c\in(1,2) p c ′ ∈ ( 1 , 2 ) B d B_d B d E : L p c ( B d ) → L s ( R d + 1 ) E:L^{p_c}(B_d)\to L^s(\mathbb{R}^{d+1}) E : L p c ( B d ) → L s ( R d + 1 ) s > p c s>p_c s > p c B d B_d B d E : L s → L s E:L^s\to L^s E : L s → L s E : L 1 → L ∞ E:L^1\to L^\infty E : L 1 → L ∞ (80) as in the proof of Corollary 9.3, with n = d + 1 n=d+1 n = d + 1
For the critical estimate write p = p c p=p_c p = p c S ( t ) = e i t Δ S(t)=e^{it\Delta} S ( t ) = e i t Δ I 0 = [ 1 / 2 , 1 ] I_0=[1/2,1] I 0 = [ 1/2 , 1 ] [51 Lemma 2.2] . Let χ ∈ C c ∞ ( B ( 0 , 2 ) ) \chi\in C^\infty_c(B(0,2)) χ ∈ C c ∞ ( B ( 0 , 2 )) B ( 0 , 1 ) B(0,1) B ( 0 , 1 ) f = P N f f=P_Nf f = P N f P N = χ ( D / N ) P_N=\chi(D/N) P N = χ ( D / N ) R d \mathbb{R}^d R d Q Q Q N N N t ∈ I 0 t\in I_0 t ∈ I 0 ∣ z ∣ ≥ 8 N |z|\ge8N ∣ z ∣ ≥ 8 N S ( t ) P N S(t)P_N S ( t ) P N C M N d ( 1 + N ∣ z ∣ ) − M C_MN^d(1+N|z|)^{-M} C M N d ( 1 + N ∣ z ∣ ) − M η = ξ / N \eta=\xi/N η = ξ / N N z − 2 t N 2 η Nz-2tN^2\eta N z − 2 t N 2 η N ∣ z ∣ / 2 N|z|/2 N ∣ z ∣/2 χ \chi χ Q Q Q 30 Q 30Q 30 Q ∣ f ∣ |f| ∣ f ∣ L p L^p L p O ( ∥ f ∥ p ) O(\|f\|_p) O ( ∥ f ∥ p ) M > d M>d M > d
Translate Q Q Q h = f 1 3 Q h=f1_{3Q} h = f 1 3 Q S ( t ) P N h = P N S ( t ) h S(t)P_Nh=P_NS(t)h S ( t ) P N h = P N S ( t ) h L 2 L^2 L 2 x x x S ( t ) h ( x − w ) S(t)h(x-w) S ( t ) h ( x − w ) χ ~ ( w ) d w \widetilde{\chi}(w)\,dw χ ( w ) d w λ = 16 d N \lambda=16\sqrt d\,N λ = 16 d N g ( y ) = h ( λ y ) g(y)=h(\lambda y) g ( y ) = h ( λ y ) h ∈ L 1 ∩ L 2 h\in L^1\cap L^2 h ∈ L 1 ∩ L 2 y = λ v y=\lambda v y = λ v
∣ S ( t ) h ( z ) ∣ = ( 4 π t ) − d / 2 λ d ∣ E g ( − λ z 2 t , λ 2 4 t ) ∣ . |S(t)h(z)|=(4\pi t)^{-d/2}\lambda^d\left|Eg\left(-\frac{\lambda z}{2t},\frac{\lambda^2}{4t}\right)\right|. ∣ S ( t ) h ( z ) ∣ = ( 4 π t ) − d /2 λ d E g ( − 2 t λ z , 4 t λ 2 ) . For z = x − w / N z=x-w/N z = x − w / N x ∈ Q x\in Q x ∈ Q t ∈ I 0 t\in I_0 t ∈ I 0 E g Eg E g C d ( N 2 + ∣ w ∣ ) C_d(N^2+|w|) C d ( N 2 + ∣ w ∣ ) ( x , t ) (x,t) ( x , t ) λ d + 2 \lambda^{d+2} λ d + 2 ∥ g ∥ p = λ − d / p ∥ h ∥ p \lVert g\rVert_p=\lambda^{-d/p}\lVert h\rVert_p ∥ g ∥ p = λ − d / p ∥ h ∥ p (83) gives
∥ S ( t ) h ( x − w / N ) ∥ L p ( Q × I 0 ) ≲ λ d − ( 2 d + 2 ) / p [ log ( 2 + N 2 + ∣ w ∣ ) ] A ∥ h ∥ p . \lVert S(t)h(x-w/N)\rVert_{L^p(Q\times I_0)}\lesssim\lambda^{d-(2d+2)/p}[\log(2+N^2+|w|)]^A\lVert h\rVert_p. ∥ S ( t ) h ( x − w / N ) ∥ L p ( Q × I 0 ) ≲ λ d − ( 2 d + 2 ) / p [ log ( 2 + N 2 + ∣ w ∣ ) ] A ∥ h ∥ p . The power of λ \lambda λ ∣ χ ˇ ( w ) ∣ d w |\check{\chi}(w)|\,dw ∣ χ ˇ ( w ) ∣ d w C [ log ( 2 + N ) ] A ∥ f ∥ L p ( 30 Q ) C[\log(2+N)]^A\lVert f\rVert_{L^p(30Q)} C [ log ( 2 + N ) ] A ∥ f ∥ L p ( 30 Q ) p p p Q Q Q 30 Q 30Q 30 Q (81) on I 0 I_0 I 0 A A A [ a / 2 , a ] [a/2,a] [ a /2 , a ] a 1 / p a^{1/p} a 1/ p N N N max ( 2 , a N ) \max(2,\sqrt{a}N) max ( 2 , a N ) p p p
For the mixed norms in (82) , choose p c < a < p p_c<a<p p c < a < p b = a b=a b = a (80) . The diagonal case of [51, Corollary 1.2 takes such an L a → L a L^a\to L^a L a → L a (82) for every p ≤ r ≤ ∞ p\le r\le\infty p ≤ r ≤ ∞ ∥ ⋅ ∥ ℓ p ≤ ∥ ⋅ ∥ ℓ 2 \lVert\cdot\rVert_{\ell^p}\le\lVert\cdot\rVert_{\ell^2} ∥ ⋅ ∥ ℓ p ≤ ∥ ⋅ ∥ ℓ 2 p > 2 p>2 p > 2
In d = 1 d=1 d = 1 ( log ( 2 + N ) ) 1 / 2 (\log(2+N))^{1/2} ( log ( 2 + N ) ) 1/2 p c = 4 p_c=4 p c = 4 I 0 I_0 I 0 1 / 4 1/4 1/4 [51 Theorem 1.4 and §4.1] . It is also the sharp power for a frequency ball: modulation moves [ − N , N ] [-N,N] [ − N , N ] [ N , 3 N ] [N,3N] [ N , 3 N ]
The Dirichlet problem on a periodic fiber
The balance law in § 4 uses a Dirichlet potential smooth up to a boundary that moves with the time variables. § 5 bounds this potential by comparing determinants on the whole fiber and on small balls inside it. We prove the required existence, comparison, and smooth dependence here; the quantitative bounds in the wavelength h h h § 5 . Alesker proved smooth Dirichlet solvability on quaternionic balls [3 Theorem 1.4] ; Zhu proved it on bounded domains in quaternionic space in the presence of a smooth subsolution [81 Theorem 1.1] . Our fibers have periodic real directions and only three real coordinates per variable. Adjoining a fourth, periodic coordinate gives the local rotations needed to estimate mixed derivatives at the boundary.
Let Λ ⊂ R m \Lambda\subset\mathbb{R}^m Λ ⊂ R m V ⊂ R 2 m V\subset\mathbb{R}^{2m} V ⊂ R 2 m Ω = ( R m / Λ ) × V \Omega=(\mathbb{R}^m/\Lambda)\times V Ω = ( R m /Λ ) × V V V V i , j i,j i , j H H H Lemma 3.1 .
Theorem A.1 (Periodic Dirichlet problem). Let m ≥ 1 m\ge1 m ≥ 1 r ∈ C ∞ ( V ‾ ) r\in C^\infty(\overline{V}) r ∈ C ∞ ( V ) V V V d r ≠ 0 dr\ne0 d r = 0 ∂ V \partial V ∂ V H ( r ) > 0 H(r)>0 H ( r ) > 0 Ω ‾ \overline{\Omega} Ω B 0 , g ∈ C ∞ ( Ω ‾ ) B_0,g\in C^\infty(\overline{\Omega}) B 0 , g ∈ C ∞ ( Ω )
H ( B 0 ) ≥ 0 , g > det H ( B 0 ) on Ω ‾ . H(B_0)\ge0,\qquad g>\det H(B_0)\quad\text{on }\overline{\Omega}. H ( B 0 ) ≥ 0 , g > det H ( B 0 ) on Ω . There is a unique smooth solution, up to the boundary, of
det H ( B 0 + u ) = g , H ( B 0 + u ) > 0 , u ∣ ∂ Ω = 0. (84) \det H(B_0+u)=g,\qquad H(B_0+u)>0,\qquad u|_{\partial\Omega}=0. \tag*{(84)} det H ( B 0 + u ) = g , H ( B 0 + u ) > 0 , u ∣ ∂ Ω = 0. (84) There are constants b ≥ b ′ > 0 b\ge b'>0 b ≥ b ′ > 0
b r ≤ u ≤ b ′ r . (85) br\le u\le b'r. \tag*{(85)} b r ≤ u ≤ b ′ r . (85) For smooth finite-dimensional families of such data and domains, these constants can be chosen locally uniformly. The solutions depend smoothly on the parameters up to the boundary, after any smooth local identification of the domains with a fixed domain.
Proof. Work on Ω ~ = Ω × ( R / Z ) m \widetilde{\Omega} = \Omega\times(\mathbb{R}/\mathbb{Z})^{m} Ω = Ω × ( R / Z ) m k k k G G G H H H H ( D 2 v ) = H ( v ) \mathcal{H}(D^{2}v) = H(v) H ( D 2 v ) = H ( v ) ξ ∈ R 4 m \xi\in\mathbb{R}^{4m} ξ ∈ R 4 m ξ ^ \hat{\xi} ξ ^ H ( ξ ξ t ) = ξ ^ ξ ^ ∗ \mathcal{H}(\xi\xi^{t}) = \hat{\xi}\hat{\xi}^{*} H ( ξ ξ t ) = ξ ^ ξ ^ ∗ tr H ( v ) = Δ v \operatorname{tr} H(v) = \Delta v tr H ( v ) = Δ v
Comparison and continuity. Join the known solution u = r u = r u = r g t = ( 1 − t ) det H ( B 0 + r ) + t g g_{t} = (1-t)\det H(B_{0}+r)+tg g t = ( 1 − t ) det H ( B 0 + r ) + t g 0 ≤ t ≤ 1 0 \leq t \leq1 0 ≤ t ≤ 1 ψ = B 0 + u \psi= B_{0}+u ψ = B 0 + u Q = H ( ψ ) Q = H(\psi) Q = H ( ψ ) S = tr Q − 1 S = \operatorname{tr} Q^{-1} S = tr Q − 1
L w = Q − 1 : H ( w ) , σ L ( ξ ) = ξ ^ ∗ Q − 1 ξ ^ . Lw = Q^{-1}:H(w), \qquad\sigma_{L}(\xi) = \hat{\xi}^{*}Q^{-1}\hat{\xi}. L w = Q − 1 : H ( w ) , σ L ( ξ ) = ξ ^ ∗ Q − 1 ξ ^ . Thus L L L Q > 0 Q > 0 Q > 0 ψ 1 , ψ 2 \psi_{1},\psi_{2} ψ 1 , ψ 2 Q i = H ( ψ i ) > 0 Q_{i} = H(\psi_{i}) > 0 Q i = H ( ψ i ) > 0
log det Q 1 − log det Q 2 = ( ∫ 0 1 ( ( 1 − θ ) Q 2 + θ Q 1 ) − 1 d θ ) : H ( ψ 1 − ψ 2 ) . \log\det Q_{1}-\log\det Q_{2} = \left(\int_{0}^{1}\left((1-\theta)Q_{2}+\theta Q_{1}\right)^{-1}\,\mathrm{d}\theta\right):H(\psi_{1}-\psi_{2}). log det Q 1 − log det Q 2 = ( ∫ 0 1 ( ( 1 − θ ) Q 2 + θ Q 1 ) − 1 d θ ) : H ( ψ 1 − ψ 2 ) . When the Hessians are positive on the closure, the averaged operator is uniformly elliptic. If det Q 1 ≥ det Q 2 \det Q_{1} \geq\det Q_{2} det Q 1 ≥ det Q 2 ψ 1 ≤ ψ 2 \psi_{1} \leq\psi_{2} ψ 1 ≤ ψ 2 ψ 1 ≤ ψ 2 \psi_{1} \leq\psi_{2} ψ 1 ≤ ψ 2 L : C 0 k + 2 , α → C k , α L:C_{0}^{k+2,\alpha}\to C^{k,\alpha} L : C 0 k + 2 , α → C k , α k ≥ 0 k\geq0 k ≥ 0 0 < α < 1 0<\alpha<1 0 < α < 1 [31 Theorems 6.14 and 6.19] apply in a finite cover of the flat quotient. In the continuity method from Δ \Delta Δ L L L r r r H ( r ) ≥ c I H(r)\geq cI H ( r ) ≥ c I C 0 C^{0} C 0 t t t
Both endpoints of the path have determinant strictly above det H ( B 0 ) \det H(B_{0}) det H ( B 0 ) 0 < b ′ < 1 < b 0<b'<1<b 0 < b ′ < 1 < b
det H ( B 0 + b ′ r ) < g t < det H ( B 0 + b r ) \det H(B_{0}+b'r)<g_{t}<\det H(B_{0}+br) det H ( B 0 + b ′ r ) < g t < det H ( B 0 + b r ) everywhere, using H ( r ) ≥ c I H(r)\geq cI H ( r ) ≥ c I u u u u = α r u=\alpha r u = α r α \alpha α b ′ ≤ α ≤ b b'\leq\alpha\leq b b ′ ≤ α ≤ b ∇ u = α ∇ r \nabla u=\alpha\nabla r ∇ u = α ∇ r L r ≥ c S Lr\geq cS L r ≥ c S S ≥ m g t − 1 / m ≥ c 0 > 0 S\geq mg_{t}^{-1/m}\geq c_{0}>0 S ≥ m g t − 1/ m ≥ c 0 > 0 e e e
L ψ e = ( log g t ) e , L ψ e e ≥ ( log g t ) e e . (86) L\psi_{e}=(\log g_{t})_{e}, \qquad L\psi_{ee}\geq(\log g_{t})_{ee}. \tag*{(86)} L ψ e = ( log g t ) e , L ψ ee ≥ ( log g t ) ee . (86) The maximum principle applied to ± ψ e + C r \pm\psi_{e}+Cr ± ψ e + C r ψ e e + C r \psi_{ee}+Cr ψ ee + C r
Second derivatives at the boundary. Center a flat chart at a boundary point, let ν \nu ν v , w v,w v , w D 2 u ( v , w ) = α D 2 r ( v , w ) D^{2}u(v,w)=\alpha D^{2}r(v,w) D 2 u ( v , w ) = α D 2 r ( v , w ) [15 Section 2, (2.11)–(2.14)] . Fix a real tangent unit vector e e e X = ( e + A x ) ⋅ ∇ X=(e+Ax)\cdot\nabla X = ( e + A x ) ⋅ ∇ X r Xr X r
v = P t a n D 2 r e ∣ ∇ r ∣ , A = v ν ∗ − ν v ∗ − ν ( v ∗ ν ) ν ∗ . v=\frac{P_{\mathrm{tan}}D^{2}r\,e}{|\nabla r|}, \qquad A=v\nu^{*}-\nu v^{*}-\nu(v^{*}\nu)\nu^{*}. v = ∣∇ r ∣ P tan D 2 r e , A = v ν ∗ − ν v ∗ − ν ( v ∗ ν ) ν ∗ . @page 39/59 starts_mid=1 ends_mid=1 where P t a n P_{\mathrm{tan}} P tan Re ( ν ∗ v ) = 0 \operatorname{Re}(\nu^*v)=0 Re ( ν ∗ v ) = 0 A ∗ = − A A^*=-A A ∗ = − A A ν = v A\nu=v A ν = v X r ( 0 ) = 0 Xr(0)=0 X r ( 0 ) = 0 w w w
∂ w ( X r ) ( 0 ) = D 2 r ( w , e ) − ∣ ∇ r ∣ w ⋅ A ν = 0. \partial_w(Xr)(0)=D^2r(w,e)-|\nabla r|\,w\cdot A\nu=0. ∂ w ( X r ) ( 0 ) = D 2 r ( w , e ) − ∣∇ r ∣ w ⋅ A ν = 0. Thus ∣ X u ∣ ≤ ∣ α X r ∣ ≤ C ∣ x ∣ 2 |Xu|\leq|\alpha Xr|\leq C|x|^2 ∣ X u ∣ ≤ ∣ α X r ∣ ≤ C ∣ x ∣ 2 X X X H H H L ( X ψ ) = X log g t L(X\psi)=X\log g_t L ( X ψ ) = X log g t Q Q Q Q − 1 Q^{-1} Q − 1 X B 0 XB_0 X B 0 ∣ L ( X u ) ∣ ≤ C S |L(Xu)|\leq CS ∣ L ( X u ) ∣ ≤ C S
In a chart ball of fixed small radius, set w = C 1 ∣ x ∣ 2 − C 2 r w=C_1|x|^2-C_2r w = C 1 ∣ x ∣ 2 − C 2 r C 1 C_1 C 1 w ≥ ∣ X u ∣ w\geq|Xu| w ≥ ∣ X u ∣ C 2 C_2 C 2 L ( w ± X u ) ≤ 0 L(w\pm Xu)\leq0 L ( w ± X u ) ≤ 0 H ( ∣ x ∣ 2 ) = 8 I H(|x|^2)=8I H ( ∣ x ∣ 2 ) = 8 I L r ≥ c S Lr\geq cS L r ≥ c S ∣ X u ∣ ≤ w |Xu|\leq w ∣ X u ∣ ≤ w ∣ ∂ ν X u ∣ |\partial_\nu Xu| ∣ ∂ ν X u ∣ ∣ D 2 u ( ν , e ) ∣ |D^2u(\nu,e)| ∣ D 2 u ( ν , e ) ∣ ∇ u ⋅ A ν \nabla u\cdot A\nu ∇ u ⋅ A ν ∇ u \nabla u ∇ u
For the remaining derivative, the product rule at the boundary reads
H ( u ) = α H ( r ) + G ( α ) G ( r ) ∗ + G ( r ) G ( α ) ∗ , G ( r ) = ∣ ∇ r ∣ ν . H(u)=\alpha H(r)+G(\alpha)G(r)^*+G(r)G(\alpha)^*,\qquad G(r)=|\nabla r|\nu. H ( u ) = α H ( r ) + G ( α ) G ( r ) ∗ + G ( r ) G ( α ) ∗ , G ( r ) = ∣∇ r ∣ ν . Compression to the quaternionic orthogonal complement of ν \nu ν Q Q Q b ′ H ( r ) ∣ ν ⊥ ≥ c I b'H(r)|_{\nu^\perp}\geq cI b ′ H ( r ) ∣ ν ⊥ ≥ c I ν \nu ν
Q = ( ψ ν ν + c 1 z ∗ z D ) . Q=\begin{pmatrix}
\psi_{\nu\nu}+c_1 & z^*\\
z & D
\end{pmatrix}. Q = ( ψ ν ν + c 1 z z ∗ D ) . Here c 1 , z , D c_1,z,D c 1 , z , D m > 1 m>1 m > 1 Lemma 3.2 gives ψ ν ν = g t / det D + z ∗ D − 1 z − c 1 \psi_{\nu\nu}=g_t/\det D+z^*D^{-1}z-c_1 ψ ν ν = g t / det D + z ∗ D − 1 z − c 1 D ≥ c I D\geq cI D ≥ c I m = 1 m=1 m = 1 Q = g t Q=g_t Q = g t D 2 ψ ≤ C I D^2\psi\leq CI D 2 ψ ≤ C I Δ ψ = tr Q > 0 \Delta\psi=\operatorname{tr}Q>0 Δ ψ = tr Q > 0 − ( 4 m − 1 ) C -(4m-1)C − ( 4 m − 1 ) C C 2 C^2 C 2
Real elliptic regularity. The C 2 C^2 C 2 det Q = g t ≥ c 0 > 0 \det Q=g_t\geq c_0>0 det Q = g t ≥ c 0 > 0 Q Q Q [ c , C ] ⊂ ( 0 , ∞ ) [c,C]\subset(0,\infty) [ c , C ] ⊂ ( 0 , ∞ ) D 2 ψ D^2\psi D 2 ψ K = { M : ∣ M ∣ ≤ C 1 , c I ≤ H ( M ) ≤ C I } K=\{M:|M|\leq C_1,\ cI\leq H(M)\leq CI\} K = { M : ∣ M ∣ ≤ C 1 , c I ≤ H ( M ) ≤ C I } H ( M ) > 0 H(M)>0 H ( M ) > 0 F ( M ) = log det H ( M ) F(M)=\log\det H(M) F ( M ) = log det H ( M ) M ∈ K M\in K M ∈ K ξ \xi ξ
D F ( M ) [ ξ ξ t ] = ξ ∗ H ( M ) − 1 ξ ∈ [ C − 1 ∣ ξ ∣ 2 , c − 1 ∣ ξ ∣ 2 ] . DF(M)[\xi\xi^t]=\xi^*H(M)^{-1}\xi\in[C^{-1}|\xi|^2,c^{-1}|\xi|^2]. D F ( M ) [ ξ ξ t ] = ξ ∗ H ( M ) − 1 ξ ∈ [ C − 1 ∣ ξ ∣ 2 , c − 1 ∣ ξ ∣ 2 ] . These are real uniform ellipticity and concavity on the range of the solution Hessians, the hypotheses (i)′, (ii)′ of the interior and boundary estimates [31 Theorems 17.14 and 17.26′] . To put F F F { H ( M ) > 0 } \{H(M)>0\} { H ( M ) > 0 } K K K F ( D 2 ψ ) − log g t = 0 F(D^2\psi)-\log g_t=0 F ( D 2 ψ ) − log g t = 0 B 0 B_0 B 0 [48 ] ) and the interior Evans–Krylov estimate [22 , 47 ] apply in a finite cover of the flat quotient, giving a uniform C 2 , α C^{2,\alpha} C 2 , α α > 0 \alpha>0 α > 0
For a limit of solvable parameters, a subsequence converges in C 2 , β C^{2,\beta} C 2 , β 0 < β < α 0<\beta<\alpha 0 < β < α g g g
Uniqueness on Ω ^ \widehat{\Omega} Ω Ω \Omega Ω C 0 k + 2 , α C^{k+2,\alpha}_0 C 0 k + 2 , α b , b ′ b,b' b , b ′
The sharp logarithm with angular regularity
Write p = p c p=p_c p = p c d = n − 1 d=n-1 d = n − 1 α = d / 2 \alpha=d/2 α = d /2 I = [ 1 , 2 ] I=[1,2] I = [ 1 , 2 ] L ( v ) = log ( 2 + v ) L(v)=\log(2+v) L ( v ) = log ( 2 + v ) α p = n \alpha p=n α p = n Λ ω = ( 1 − Δ S n − 1 ) 1 / 2 \Lambda_\omega=(1-\Delta_{S^{n-1}})^{1/2} Λ ω = ( 1 − Δ S n − 1 ) 1/2 ∥ u ∥ p , ∞ = sup λ > 0 λ ∣ { ∣ u ∣ > λ } ∣ 1 / p \|u\|_{p,\infty}=\sup_{\lambda>0}\lambda|\{|u|>\lambda\}|^{1/p} ∥ u ∥ p , ∞ = sup λ > 0 λ ∣ { ∣ u ∣ > λ } ∣ 1/ p
Theorem B.1 (The angular endpoint). Let n ≥ 2 n\ge2 n ≥ 2 p = p c p=p_c p = p c B > 1 / 2 + 1 / p B>1/2+1/p B > 1/2 + 1/ p
∥ U ( T ) f ∥ L p ( R n × I ) ≤ C n , B ( ∥ L ( J ) 1 / p f ∥ p + ∥ L ( Λ ω ) B f ∥ p ) , \|U(T)f\|_{L^p(\mathbb{R}^n\times I)}\le C_{n,B}\bigl(\|L(J)^{1/p}f\|_p+\|L(\Lambda_\omega)^Bf\|_p\bigr), ∥ U ( T ) f ∥ L p ( R n × I ) ≤ C n , B ( ∥ L ( J ) 1/ p f ∥ p + ∥ L ( Λ ω ) B f ∥ p ) , ∥ U ( T ) f ∥ L p , ∞ ( R n × I ) ≤ C n , B ∥ L ( Λ ω ) B f ∥ p . \|U(T)f\|_{L^{p,\infty}(\mathbb{R}^n\times I)}\le C_{n,B}\|L(\Lambda_\omega)^Bf\|_p. ∥ U ( T ) f ∥ L p , ∞ ( R n × I ) ≤ C n , B ∥ L ( Λ ω ) B f ∥ p . In each inequality the angular norm on the right can be replaced by ∥ Λ ω σ f ∥ p \|\Lambda_\omega^\sigma f\|_p ∥ Λ ω σ f ∥ p σ > 0 \sigma>0 σ > 0 σ \sigma σ
For radial data the angular operator acts as a scalar, so Proposition 8.1 makes the frequency power 1 / p 1/p 1/ p Π N = χ ( ∣ D ∣ / N ) \Pi_N=\chi(|D|/N) Π N = χ ( ∣ D ∣/ N ) χ ∈ C 0 ∞ ( ( 0 , ∞ ) ) \chi\in C^\infty_0((0,\infty)) χ ∈ C 0 ∞ (( 0 , ∞ )) A = 2 / p A=2/p A = 2/ p A A A M M M M M M
Proposition B.2 (Bounded angular degree on a frequency annulus). For N ≥ 2 N\ge2 N ≥ 2 f ∈ L p ( R n ) f\in L^p(\mathbb{R}^n) f ∈ L p ( R n ) M M M
∥ U ( t ) Π N f ∥ L p ( I × R n ) ≲ [ L ( N ) 1 / p + L ( M ) A ] ∥ f ∥ p , (87) \|U(t)\Pi_Nf\|_{L^p(I\times\mathbb{R}^n)}\lesssim[L(N)^{1/p}+L(M)^A]\|f\|_p, \tag*{(87)} ∥ U ( t ) Π N f ∥ L p ( I × R n ) ≲ [ L ( N ) 1/ p + L ( M ) A ] ∥ f ∥ p , (87) ∥ U ( t ) Π N f ∥ L p , ∞ ( I × R n ) ≲ L ( M ) A ∥ f ∥ p . \|U(t)\Pi_Nf\|_{L^{p,\infty}(I\times\mathbb{R}^n)}\lesssim L(M)^A\|f\|_p. ∥ U ( t ) Π N f ∥ L p , ∞ ( I × R n ) ≲ L ( M ) A ∥ f ∥ p . The constants are independent of M M M N N N
The two terms in the strong estimate come from different output radii in physical space. When M M M N N N ( 2 + M ) K / N (2+M)^K/N ( 2 + M ) K / N K K K ( 2 + M ) K (2+M)^K ( 2 + M ) K O ( L ( M ) A ) O(L(M)^A) O ( L ( M ) A ) x = r θ x=r\theta x = r θ r ≤ 4 r\le4 r ≤ 4 r − α G ( t ± r , θ ) r^{-\alpha}G(t\pm r,\theta) r − α G ( t ± r , θ ) ∥ G ∥ L p ( d v d θ ) ≲ ∥ f ∥ p \|G\|_{L^p(dv\,d\theta)}\lesssim\|f\|_p ∥ G ∥ L p ( d v d θ ) ≲ ∥ f ∥ p v = t ± r v=t\pm r v = t ± r r r r p p p r d − α p = r − 1 r^{d-\alpha p}=r^{-1} r d − α p = r − 1 L p L^p L p r d d r r^d\,dr r d d r
For radial data, Müller and Seeger reduced the wave kernel to four one-dimensional kernels concentrated at t ± r ± s = 0 t\pm r\pm s=0 t ± r ± s = 0 s s s [58 Lemma 2.1] . Colzani, Cominardi and Stepan obtained the corresponding weak endpoint for the radial cosine propagator [18 Theorem 2.1] . We recover these four kernels outside the indicated ball with constants uniform in the angular degree.
Observation on a ball
Lemma B.3. For N ≥ 2 N \ge2 N ≥ 2 0 < r ≤ 1 0 < r \le1 0 < r ≤ 1 x 0 ∈ R n x_0 \in\mathbb{R}^n x 0 ∈ R n
∥ U ( t ) Π N f ∥ L p ( I × B ( x 0 , r ) ) ≲ min { 1 , ( N r ) n / p } L ( N r ) A ∥ f ∥ p . \|U(t)\Pi_N f\|_{L^p(I \times B(x_0,r))} \lesssim\min\{1,(Nr)^{n/p}\}L(Nr)^A\|f\|_p. ∥ U ( t ) Π N f ∥ L p ( I × B ( x 0 , r )) ≲ min { 1 , ( N r ) n / p } L ( N r ) A ∥ f ∥ p . Proof. Translate to x 0 = 0 x_0=0 x 0 = 0 T = N T=N T = N R = N r R=Nr R = N r [ T , 2 T ] × B ( 0 , R ) [T,2T]\times B(0,R) [ T , 2 T ] × B ( 0 , R ) T 1 / p min ( 1 , R n / p ) L ( R ) A ∥ f ∥ p T^{1/p}\min(1,R^{n/p})L(R)^A\|f\|_p T 1/ p min ( 1 , R n / p ) L ( R ) A ∥ f ∥ p U ( t ) Π 1 U(t)\Pi_1 U ( t ) Π 1 v = ∣ z ∣ v=|z| v = ∣ z ∣ ⟨ v ⟩ = ( 1 + v 2 ) 1 / 2 \langle v\rangle=(1+v^2)^{1/2} ⟨ v ⟩ = ( 1 + v 2 ) 1/2 D D D v ≥ 1 v\ge1 v ≥ 1
K ( t , z ) = v − α k ( t − v ) + O D ( v − α − 1 ⟨ t − v ⟩ − D + v − α ⟨ t + v ⟩ − D ) . (88) K(t,z)=v^{-\alpha}k(t-v)+O_D\left(v^{-\alpha-1}\langle t-v\rangle^{-D}+v^{-\alpha}\langle t+v\rangle^{-D}\right). \tag*{(88)} K ( t , z ) = v − α k ( t − v ) + O D ( v − α − 1 ⟨ t − v ⟩ − D + v − α ⟨ t + v ⟩ − D ) . (88) Here k k k ρ α χ ( ρ ) \rho^\alpha\chi(\rho) ρ α χ ( ρ ) e ± i v ρ e^{\pm iv\rho} e ± i v ρ v ρ v\rho v ρ v < 1 v<1 v < 1 O D ( ⟨ t ⟩ − D ) O_D(\langle t\rangle^{-D}) O D (⟨ t ⟩ − D ) x x x T / 2 ≤ s = ∣ y − x ∣ ≤ 3 T T/2\le s=|y-x|\le3T T /2 ≤ s = ∣ y − x ∣ ≤ 3 T
∥ U ( t ) Π 1 f ( x ) ∥ L p ( [ T , 2 T ] ) ≲ T 1 / p ∥ f ∥ p . \|U(t)\Pi_1f(x)\|_{L^p([T,2T])}\lesssim T^{1/p}\|f\|_p. ∥ U ( t ) Π 1 f ( x ) ∥ L p ([ T , 2 T ]) ≲ T 1/ p ∥ f ∥ p . Indeed the convolution acts on s α ∫ S d − 1 ∣ f ( x + s ω ) ∣ d ω s^\alpha\int_{\mathbb{S}^{d-1}}|f(x+s\omega)|\,d\omega s α ∫ S d − 1 ∣ f ( x + s ω ) ∣ d ω p p p C T ∥ f ∥ p p CT\|f\|_p^p C T ∥ f ∥ p p α p = d + 1 \alpha p=d+1 α p = d + 1 O D ( ( T + s ) − D ) O_D((T+s)^{-D}) O D (( T + s ) − D ) x x x R ≤ 2 R\le2 R ≤ 2
Suppose 2 < R ≤ T 2<R\le T 2 < R ≤ T S = 100 R 3 + n / p S=100R^{3+n/p} S = 100 R 3 + n / p T ≤ 10 S T\le10S T ≤ 10 S Theorem 1.2 suffices since L ( T ) ≲ L ( R ) L(T)\lesssim L(R) L ( T ) ≲ L ( R ) T > 10 S T>10S T > 10 S Q Q Q Q Q Q O ( R ) O(R) O ( R ) S S S L ( R ) A L(R)^A L ( R ) A Q ≥ S Q\ge S Q ≥ S
f Q ( s ω ) = s − α h ( s − Q , ω ) , h ( u , ω ) = 0 ( ∣ u ∣ > 5 R ) . f_Q(s\omega)=s^{-\alpha}h(s-Q,\omega),\qquad h(u,\omega)=0\quad(|u|>5R). f Q ( s ω ) = s − α h ( s − Q , ω ) , h ( u , ω ) = 0 ( ∣ u ∣ > 5 R ) . Its contribution at t = Q + b t=Q+b t = Q + b ∣ x ∣ , ∣ b ∣ ≤ R |x|,|b|\le R ∣ x ∣ , ∣ b ∣ ≤ R
T h ( b , x ) = ∫ k ( b − u + x ⋅ ω ) h ( u , ω ) d u d ω . \mathcal{T}h(b,x)=\int k(b-u+x\cdot\omega)h(u,\omega)\,du\,d\omega. T h ( b , x ) = ∫ k ( b − u + x ⋅ ω ) h ( u , ω ) d u d ω . This expression is independent of Q Q Q ∣ s ω − x ∣ = s − x ⋅ ω + O ( R 2 / Q ) |s\omega-x|=s-x\cdot\omega+O(R^2/Q) ∣ s ω − x ∣ = s − x ⋅ ω + O ( R 2 / Q ) ( s / ∣ s ω − x ∣ ) α = 1 + O ( R / Q ) (s/|s\omega-x|)^\alpha=1+O(R/Q) ( s /∣ s ω − x ∣ ) α = 1 + O ( R / Q ) s d s^d s d h h h C D ( R 2 / Q ) ⟨ b − u + x ⋅ ω ⟩ − D C_D(R^2/Q)\langle b-u+x\cdot\omega\rangle^{-D} C D ( R 2 / Q ) ⟨ b − u + x ⋅ ω ⟩ − D b b b ω \omega ω x x x
∥ U ( Q + b ) Π 1 f Q − T h ∥ L p ( { ∣ b ∣ , ∣ x ∣ ≤ R } ) ≲ ( R 2 + n / p / Q ) ∥ h ∥ L p ( d u d ω ) . \|U(Q+b)\Pi_1f_Q-\mathcal{T}h\|_{L^p(\{|b|,|x|\le R\})}\lesssim(R^{2+n/p}/Q)\|h\|_{L^p(du\,d\omega)}. ∥ U ( Q + b ) Π 1 f Q − T h ∥ L p ({ ∣ b ∣ , ∣ x ∣ ≤ R }) ≲ ( R 2 + n / p / Q ) ∥ h ∥ L p ( d u d ω ) . We may therefore estimate the profile at the smaller radius Q = S Q=S Q = S d − α p = − 1 d-\alpha p=-1 d − α p = − 1
∥ f S ∥ p p = ∫ ∣ h ( u , ω ) ∣ p S + u d u d ω ≍ S − 1 ∥ h ∥ p p . \|f_S\|_p^p=\int\frac{|h(u,\omega)|^p}{S+u}\,du\,d\omega\asymp S^{-1}\|h\|_p^p. ∥ f S ∥ p p = ∫ S + u ∣ h ( u , ω ) ∣ p d u d ω ≍ S − 1 ∥ h ∥ p p . The factor S 1 / p S^{1/p} S 1/ p Theorem 1.2 on [ S / 2 , S ] [S/2,S] [ S /2 , S ] [ S , 2 S ] [S,2S] [ S , 2 S ] L ( S ) ≲ L ( R ) L(S)\lesssim L(R) L ( S ) ≲ L ( R ) ∥ T h ∥ L p ( { ∣ b ∣ , ∣ x ∣ ≤ R } ) ≲ L ( R ) A ∥ h ∥ p \|\mathcal{T}h\|_{L^p(\{|b|,|x|\le R\})}\lesssim L(R)^A\|h\|_p ∥ T h ∥ L p ({ ∣ b ∣ , ∣ x ∣ ≤ R }) ≲ L ( R ) A ∥ h ∥ p
Divide [ T , 2 T ] [T,2T] [ T , 2 T ] R / 2 R/2 R /2 R R R Q i Q_i Q i i i i h i ( u , ω ) = ( Q i + u ) α f ( ( Q i + u ) ω ) 1 ∣ u ∣ ≤ 5 R h_i(u,\omega)=(Q_i+u)^\alpha f((Q_i+u)\omega)1_{|u|\le5R} h i ( u , ω ) = ( Q i + u ) α f (( Q i + u ) ω ) 1 ∣ u ∣ ≤ 5 R ∑ i ∥ h i ∥ p p ≲ T ∥ f ∥ p p \sum_i\|h_i\|_p^p\lesssim T\|f\|_p^p ∑ i ∥ h i ∥ p p ≲ T ∥ f ∥ p p T / 2 ≤ s ≤ 3 T T/2\le s\le3T T /2 ≤ s ≤ 3 T ∣ s − Q i ∣ > 5 R |s-Q_i|>5R ∣ s − Q i ∣ > 5 R C D s − α 1 ∣ t − s ∣ > 3 R ⟨ t − s ⟩ − D C_Ds^{-\alpha}1_{|t-s|>3R}\langle t-s\rangle^{-D} C D s − α 1 ∣ t − s ∣ > 3 R ⟨ t − s ⟩ − D R R R t t t C D R n / p + 1 − D T 1 / p ∥ f ∥ p C_DR^{n/p+1-D}T^{1/p}\|f\|_p C D R n / p + 1 − D T 1/ p ∥ f ∥ p [ T / 2 , 3 T ] [T/2,3T] [ T /2 , 3 T ] T + s T+s T + s
The polar kernel and the two endpoint norms
At an output point r θ r\theta r θ θ \theta θ − θ -\theta − θ g g g R n \mathbb{R}^{n} R n v ∈ R v \in\mathbb{R} v ∈ R
h g ( v , θ ) = ∣ v ∣ d / p ( ∣ g ( ∣ v ∣ θ ) ∣ + ∣ g ( − ∣ v ∣ θ ) ∣ + ∫ S d ∣ g ( ∣ v ∣ ω ) ∣ d ω ) , G g ( v , θ ) = M R h g ( v , θ ) , h_g(v,\theta)=|v|^{d/p}\left(|g(|v|\theta)|+|g(-|v|\theta)|+\int_{\mathbb{S}^{d}}|g(|v|\omega)|\,\mathrm{d}\omega\right), \qquad G_g(v,\theta)=\mathcal{M}_{\mathbb{R}}h_g(v,\theta), h g ( v , θ ) = ∣ v ∣ d / p ( ∣ g ( ∣ v ∣ θ ) ∣ + ∣ g ( − ∣ v ∣ θ ) ∣ + ∫ S d ∣ g ( ∣ v ∣ ω ) ∣ d ω ) , G g ( v , θ ) = M R h g ( v , θ ) , where M R \mathcal{M}_{\mathbb{R}} M R v v v ∣ v ∣ d / p |v|^{d/p} ∣ v ∣ d / p ∥ G g ∥ L p ( d v d θ ) ≲ ∥ g ∥ p \|G_g\|_{L^p(\mathrm{d}v\,\mathrm{d}\theta)}\lesssim\|g\|_p ∥ G g ∥ L p ( d v d θ ) ≲ ∥ g ∥ p
Lemma B.4. There is an integer K K K M , N M,N M , N g g g M M M s = ∣ y ∣ ≥ s 0 s=|y|\ge s_0 s = ∣ y ∣ ≥ s 0 N min ( r , s 0 ) ≥ ( 2 + M ) K N\min(r,s_0)\ge(2+M)^K N min ( r , s 0 ) ≥ ( 2 + M ) K t ∈ I t\in I t ∈ I
∣ U ( t ) Π N g ( r θ ) ∣ ≲ ∑ ± G g ( t ± r , θ ) { r − α , r ≤ 4 , r − d / p , r > 4. (89) |U(t)\Pi_Ng(r\theta)|\lesssim\sum_{\pm}G_g(t\pm r,\theta)
\begin{cases}
r^{-\alpha}, & r\le4,\\
r^{-d/p}, & r>4.
\end{cases} \tag*{(89)} ∣ U ( t ) Π N g ( r θ ) ∣ ≲ ± ∑ G g ( t ± r , θ ) { r − α , r − d / p , r ≤ 4 , r > 4. (89) For arbitrary g ∈ L p ( R n ) g\in L^p(\mathbb{R}^n) g ∈ L p ( R n ) ∣ y ∣ < N η − 1 |y|<N^{\eta-1} ∣ y ∣ < N η − 1 η = α / ( 2 n ) \eta=\alpha/(2n) η = α / ( 2 n )
∥ U ( t ) Π N ( 1 ∣ y ∣ < N η − 1 g ) ∥ L p ( I × R n ) ≲ N − δ ∥ g ∥ p , δ = ( α / 2 ) ( 1 − 2 / p ) > 0. (90) \|U(t)\Pi_N(\mathbf{1}_{|y|<N^{\eta-1}}g)\|_{L^p(I\times\mathbb{R}^n)}\lesssim N^{-\delta}\|g\|_p,\qquad\delta=(\alpha/2)(1-2/p)>0. \tag*{(90)} ∥ U ( t ) Π N ( 1 ∣ y ∣ < N η − 1 g ) ∥ L p ( I × R n ) ≲ N − δ ∥ g ∥ p , δ = ( α /2 ) ( 1 − 2/ p ) > 0. (90) Proof. Write S z F ( θ ) = ∫ S d e i z θ ⋅ ω F ( ω ) d ω S_zF(\theta)=\int_{\mathbb{S}^{d}}e^{iz\theta\cdot\omega}F(\omega)\,\mathrm{d}\omega S z F ( θ ) = ∫ S d e i z θ ⋅ ω F ( ω ) d ω r , s r,s r , s
c n s d ∫ e − i t ρ χ ( ρ / N ) ρ d S r ρ S − s ρ d ρ . c_ns^d\int e^{-it\rho}\chi(\rho/N)\rho^dS_{r\rho}S_{-s\rho}\,\mathrm{d}\rho. c n s d ∫ e − i tρ χ ( ρ / N ) ρ d S r ρ S − s ρ d ρ . For each sign of z z z
S z F = ∣ z ∣ − α ∑ σ = ± 1 e i σ ∣ z ∣ ( c σ P σ F + E σ , z F ) . S_zF=|z|^{-\alpha}\sum_{\sigma=\pm1}e^{i\sigma|z|}(c_\sigma P_\sigma F+E_{\sigma,z}F). S z F = ∣ z ∣ − α σ = ± 1 ∑ e iσ ∣ z ∣ ( c σ P σ F + E σ , z F ) . where for either fixed sign of z z z P σ P_\sigma P σ D > 2 + α D>2+\alpha D > 2 + α E σ , z F E_{\sigma,z}F E σ , z F D D D ∣ z ∣ |z| ∣ z ∣ C D ∣ z ∣ − 1 ∥ F ∥ C m D C_D|z|^{-1}\|F\|_{C^{m_D}} C D ∣ z ∣ − 1 ∥ F ∥ C m D 1 ∓ θ ⋅ ω = ∣ w ∣ 2 / 2 1\mp\theta\cdot\omega=|w|^2/2 1 ∓ θ ⋅ ω = ∣ w ∣ 2 /2 I λ ( a ) = λ d / 2 ∫ e ± i λ ∣ w ∣ 2 / 2 a ( w ) d w I_\lambda(a)=\lambda^{d/2}\int e^{\pm i\lambda|w|^2/2}a(w)\,\mathrm{d}w I λ ( a ) = λ d /2 ∫ e ± iλ ∣ w ∣ 2 /2 a ( w ) d w λ = ∣ z ∣ \lambda=|z| λ = ∣ z ∣ w w w λ ∂ λ I λ ( a ) = − 1 2 I λ ( w ⋅ ∇ a ) \lambda\partial_\lambda I_\lambda(a)=-\frac12 I_\lambda(w\cdot\nabla a) λ ∂ λ I λ ( a ) = − 2 1 I λ ( w ⋅ ∇ a ) O ( λ − 1 ) O(\lambda^{-1}) O ( λ − 1 ) [74 Chapter VIII, §2] .
On degrees at most M M M ∥ F ∥ 2 ≲ ( 2 + M ) d / 2 ∥ F ∥ 1 \|F\|_2\lesssim(2+M)^{d/2}\|F\|_1 ∥ F ∥ 2 ≲ ( 2 + M ) d /2 ∥ F ∥ 1 − Δ S d -\Delta_{\mathbb{S}^d} − Δ S d ∥ F ∥ C m D ≲ ( 2 + M ) c D ∥ F ∥ 1 \|F\|_{C^{m_D}}\lesssim(2+M)^{c_D}\|F\|_1 ∥ F ∥ C m D ≲ ( 2 + M ) c D ∥ F ∥ 1 K > c D K>c_D K > c D D D D ∣ z ∣ ≳ ( 2 + M ) K |z|\gtrsim(2+M)^K ∣ z ∣ ≳ ( 2 + M ) K L 1 L^1 L 1 L ∞ L^\infty L ∞ θ \theta θ C ( ∣ F ( θ ) ∣ + ∣ F ( − θ ) ∣ + ∥ F ∥ 1 ) C(|F(\theta)|+|F(-\theta)|+\|F\|_1) C ( ∣ F ( θ ) ∣ + ∣ F ( − θ ) ∣ + ∥ F ∥ 1 )
Now integrate by parts in ρ / N \rho/N ρ / N C D r − α s q N ⟨ N ( t ± r ± s ) ⟩ − D C_Dr^{-\alpha}s^qN\langle N(t\pm r\pm s)\rangle^{-D} C D r − α s q N ⟨ N ( t ± r ± s ) ⟩ − D s d / p s^{d/p} s d / p s 1 / p s^{1/p} s 1/ p r ≤ 4 r\le4 r ≤ 4 C ⟨ t ± r ± s ⟩ 1 / p C\langle t\pm r\pm s\rangle^{1/p} C ⟨ t ± r ± s ⟩ 1/ p r > 4 r>4 r > 4 r 1 / p r^{1/p} r 1/ p v = t ± r v=t\pm r v = t ± r v − s v-s v − s v + s v+s v + s R \mathbb{R} R h g h_g h g s s s C D N ⟨ N ⋅ ⟩ − D + 1 / p C_DN\langle N\cdot\rangle^{-D+1/p} C D N ⟨ N ⋅ ⟩ − D + 1/ p C G g ( v , θ ) CG_g(v,\theta) C G g ( v , θ )
Finally (B.2) and scaling give ∥ U ( t ) Π N ∥ 1 → ∞ ≲ N α + 1 \|U(t)\Pi_N\|_{1\to\infty}\lesssim N^{\alpha+1} ∥ U ( t ) Π N ∥ 1 → ∞ ≲ N α + 1 t ∈ I t\in I t ∈ I L 2 L^2 L 2 v = g 1 { ∣ y ∣ < b } v=g1_{\{|y|<b\}} v = g 1 { ∣ y ∣ < b } ∥ U ( t ) Π N v ∥ 2 \|U(t)\Pi_Nv\|_2 ∥ U ( t ) Π N v ∥ 2 ∥ U ( t ) Π N v ∥ ∞ \|U(t)\Pi_Nv\|_\infty ∥ U ( t ) Π N v ∥ ∞ ∥ U ( t ) Π N v ∥ p ≲ ( N α + 1 b 1 − 2 / p ) ∥ g ∥ p \|U(t)\Pi_Nv\|_p\lesssim(N^{\alpha+1}b^{1-2/p})\|g\|_p ∥ U ( t ) Π N v ∥ p ≲ ( N α + 1 b 1 − 2/ p ) ∥ g ∥ p b = N η − 1 b=N^{\eta-1} b = N η − 1
Proof of Proposition B.2. Put H = ( 2 + M ) K H=(2+M)^K H = ( 2 + M ) K H > N η H>N^\eta H > N η L ( N ) ≲ L ( M ) L(N)\lesssim L(M) L ( N ) ≲ L ( M ) H ≤ N η H\leq N^\eta H ≤ N η r ≤ H / N r\leq H/N r ≤ H / N C L ( H ) A ∥ f ∥ p ≲ L ( M ) A ∥ f ∥ p CL(H)^A\|f\|_p\lesssim L(M)^A\|f\|_p C L ( H ) A ∥ f ∥ p ≲ L ( M ) A ∥ f ∥ p s < N η − 1 s<N^{\eta-1} s < N η − 1 g g g N r Nr N r N s Ns N s H H H
Consider one profile in (B.3). Changing variables from ( t , r ) (t,r) ( t , r ) ( v , r ) = ( t ± r , r ) (v,r)=(t\pm r,r) ( v , r ) = ( t ± r , r ) v , θ v,\theta v , θ G = G g ( v , θ ) G=G_g(v,\theta) G = G g ( v , θ ) N − 1 ≤ r ≤ 4 N^{-1}\leq r\leq4 N − 1 ≤ r ≤ 4
∫ N − 1 4 ( r − α G ) p r d d r = G p log ( 4 N ) , ∫ { r > 0 : r − α G > λ } r d d r = G p ( d + 1 ) λ p ( λ > 0 ) . (91) \int_{N^{-1}}^4 (r^{-\alpha}G)^p r^d\,\mathrm{d}r=G^p\log(4N),\qquad\int_{\{r>0:\,r^{-\alpha}G>\lambda\}}r^d\,\mathrm{d}r=\frac{G^p}{(d+1)\lambda^p}\quad(\lambda>0). \tag*{(91)} ∫ N − 1 4 ( r − α G ) p r d d r = G p log ( 4 N ) , ∫ { r > 0 : r − α G > λ } r d d r = ( d + 1 ) λ p G p ( λ > 0 ) . (91) Integrating in v , θ v,\theta v , θ O ( L ( N ) 1 / p ) O(L(N)^{1/p}) O ( L ( N ) 1/ p ) r > 4 r>4 r > 4 p p p r − d / p r^{-d/p} r − d / p t t t ∣ G g ( t ± r , θ ) ∣ p |G_g(t\pm r,\theta)|^p ∣ G g ( t ± r , θ ) ∣ p r , θ r,\theta r , θ ∥ G g ∥ p p \|G_g\|_p^p ∥ G g ∥ p p N N N
For A = 2 / p A=2/p A = 2/ p M ≤ exp ( L ( N ) ) M\leq\exp(\sqrt{L(N)}) M ≤ exp ( L ( N ) ) L ( N ) 1 / p L(N)^{1/p} L ( N ) 1/ p [ 5 / 4 , 7 / 4 ] [5/4,7/4] [ 5/4 , 7/4 ]
Summation over the spatial and angular frequencies
Choose a smooth radial Littlewood–Paley decomposition f j f_j f j N j = 2 j N_j=2^j N j = 2 j u j = U ( t ) f j u_j=U(t)f_j u j = U ( t ) f j Π N j \Pi_{N_j} Π N j f ^ j \widehat f_j f j L p L^p L p s ≥ N j η − 1 s\geq N_j^{\eta-1} s ≥ N j η − 1 O ( ∥ f ∥ p ) O(\|f\|_p) O ( ∥ f ∥ p ) j j j
We group angular degrees according to the size of their logarithm. Use smooth angular low pass operators S m S_m S m M m = 2 2 m M_m=2^{2^m} M m = 2 2 m m ≥ 0 m\geq0 m ≥ 0 Q m = S m − S m − 1 Q_m=S_m-S_{m-1} Q m = S m − S m − 1 S − 1 = 0 S_{-1}=0 S − 1 = 0 M m M_m M m 2 M m 2M_m 2 M m ∣ S m ( θ , ω ) ∣ ≲ D M m d ( 1 + M m dist ( θ , ω ) ) − D |S_m(\theta,\omega)|\lesssim_D M_m^d(1+M_m\operatorname{dist}(\theta,\omega))^{-D} ∣ S m ( θ , ω ) ∣ ≲ D M m d ( 1 + M m dist ( θ , ω ) ) − D [60 Theorem 3.5 and Corollaries 3.6, 5.3] ; on the circle this is ordinary smooth Fourier summation. In particular ∣ S m g ∣ ≲ M m S d g |S_mg|\lesssim M_m S_{d}g ∣ S m g ∣ ≲ M m S d g m m m B > 0 B>0 B > 0
∥ Q m f ∥ p ≲ B 2 − m B ∥ L ( Λ ω ) B f ∥ p . (92) \|Q_m f\|_p\lesssim B\,2^{-mB}\|L(\Lambda_\omega)^Bf\|_p. \tag*{(92)} ∥ Q m f ∥ p ≲ B 2 − m B ∥ L ( Λ ω ) B f ∥ p . (92) Indeed, as a function of the harmonic degree ν \nu ν Q m Q_m Q m L ( ( 1 + ν ( ν + d − 1 ) ) 1 / 2 ) ≳ 2 m L((1+\nu(\nu+d-1))^{1/2})\gtrsim2^m L (( 1 + ν ( ν + d − 1 ) ) 1/2 ) ≳ 2 m 2 m B Q m ( ν ) L ( ( 1 + ν ( ν + d − 1 ) ) 1 / 2 ) − B 2^{mB}Q_m(\nu)L((1+\nu(\nu+d-1))^{1/2})^{-B} 2 m B Q m ( ν ) L (( 1 + ν ( ν + d − 1 ) ) 1/2 ) − B ν \nu ν [11 Theorem 4.9(1)] gives (B.6) on each angular slice.
To pay the spatial logarithm only once, at fixed r r r N j N_j N j H m = ( 2 + 2 M m ) K H_m=(2+2M_m)^K H m = ( 2 + 2 M m ) K m m m
H m ≤ min ( N j η , N j r ) . (93) H_m\leq\min(N_j^\eta,N_jr). \tag*{(93)} H m ≤ min ( N j η , N j r ) . (93) Call this index m ( j , r ) m(j,r) m ( j , r ) − 1 -1 − 1 m m m a j = 1 s ≥ N j η − 1 f j a_j=1_{s\geq N_j^{\eta-1}}f_j a j = 1 s ≥ N j η − 1 f j ( t , r θ ) (t,r\theta) ( t , r θ )
U ( t ) ∏ N j a j = U ( t ) ∏ N j S m ( j , r ) a j + ∑ m > m ( j , r ) U ( t ) ∏ N j Q m a j . U(t)\prod_{N_j}a_j=U(t)\prod_{N_j}S_{m(j,r)}a_j+\sum_{m>m(j,r)}U(t)\prod_{N_j}Q_ma_j. U ( t ) N j ∏ a j = U ( t ) N j ∏ S m ( j , r ) a j + m > m ( j , r ) ∑ U ( t ) N j ∏ Q m a j . For a given output r r r m ( j , r ) m(j,r) m ( j , r ) M S d f j M_{\mathbb{S}^d}f_j M S d f j S m ( j , r ) f j S_{m(j,r)}f_j S m ( j , r ) f j [33 Theorem 1.2] , gives for arbitrary nonnegative coefficients c j c_j c j
∥ ( c j G M S d f j ) j ∥ L p ( d v d θ ; ℓ 2 ) ≲ ∥ ( c j f j ) j ∥ L p ( R n ; ℓ 2 ) . \left\|\left(c_jG_{M_{\mathbb{S}^d}f_j}\right)_j\right\|_{L^p(dv\,d\theta;\ell^2)}
\lesssim\left\|\left(c_jf_j\right)_j\right\|_{L^p(\mathbb{R}^n;\ell^2)}. ( c j G M S d f j ) j L p ( d v d θ ; ℓ 2 ) ≲ ( c j f j ) j L p ( R n ; ℓ 2 ) . The reflection and angular integral in h g h_g h g L p ( S d ; ℓ 2 ) L^p(\mathbb{S}^d;\ell^2) L p ( S d ; ℓ 2 )
It remains to integrate these profiles along v = t ± r v=t\pm r v = t ± r j j j G j = G M S d f j ( v , θ ) G_j=G_{M_{\mathbb{S}^d}f_j}(v,\theta) G j = G M S d f j ( v , θ ) r ≥ N j − 1 r\geq N_j^{-1} r ≥ N j − 1 r ≤ 4 r\leq4 r ≤ 4 r − α r^{-\alpha} r − α C ( r + N j − 1 ) − α C(r+N_j^{-1})^{-\alpha} C ( r + N j − 1 ) − α v , θ v,\theta v , θ L p / 2 ( r d d r ) L^{p/2}(r^d\,dr) L p /2 ( r d d r )
∥ ( ∑ j ( r + N j − 1 ) − 2 α ∣ G j ∣ 2 ) 1 / 2 ∥ L p ( ( 0 , 4 ) , r d d r ) ≲ ( ∑ j ( 1 + j ) 2 / p ∣ G j ∣ 2 ) 1 / 2 . (94) \left\|\left(\sum_j(r+N_j^{-1})^{-2\alpha}|G_j|^2\right)^{1/2}\right\|_{L^p((0,4),r^d\,dr)}
\lesssim
\left(\sum_j(1+j)^{2/p}|G_j|^2\right)^{1/2}. \tag*{(94)} ( j ∑ ( r + N j − 1 ) − 2 α ∣ G j ∣ 2 ) 1/2 L p (( 0 , 4 ) , r d d r ) ≲ ( j ∑ ( 1 + j ) 2/ p ∣ G j ∣ 2 ) 1/2 . (94) Indeed the p p p j j j O ( 1 + j ) O(1+j) O ( 1 + j ) α p = n \alpha p=n α p = n v , θ v,\theta v , θ c j = ( 1 + j ) 1 / p c_j=(1+j)^{1/p} c j = ( 1 + j ) 1/ p C ∥ L ( J ) 1 / p f ∥ p C\|L(J)^{1/p}f\|_p C ∥ L ( J ) 1/ p f ∥ p r − α r^{-\alpha} r − α G = ∥ ( G j ) ∥ ℓ 2 G=\|(G_j)\|_{\ell^2} G = ∥ ( G j ) ∥ ℓ 2 c j = 1 c_j=1 c j = 1 C ∥ f ∥ p C\|f\|_p C ∥ f ∥ p r > 4 r>4 r > 4 r − d / p r^{-d/p} r − d / p t t t C ∥ f ∥ p C\|f\|_p C ∥ f ∥ p
It remains to estimate the unselected angular frequencies in strong L p L^p L p m m m v j v_j v j U ( t ) ∏ N j a U(t)\prod_{N_j}a U ( t ) ∏ N j a Q m f j 1 s ≥ N j η − 1 Q_mf_j1_{s\geq N_j^{\eta-1}} Q m f j 1 s ≥ N j η − 1 b j = ∥ Q m f j ∥ p b_j=\|Q_mf_j\|_p b j = ∥ Q m f j ∥ p H m > N j η H_m>N_j^\eta H m > N j η v j v_j v j O ( 1 + log H m ) O(1+\log H_m) O ( 1 + log H m ) j j j C L ( H m ) A b j CL(H_m)^A b_j C L ( H m ) A b j ∥ w ∥ ℓ 2 ≤ ( # j ) 1 / 2 − 1 / p ∥ w ∥ ℓ p \|w\|_{\ell^2}\leq(\#j)^{1/2-1/p}\|w\|_{\ell^p} ∥ w ∥ ℓ 2 ≤ ( # j ) 1/2 − 1/ p ∥ w ∥ ℓ p C L ( H m ) A + 1 / 2 − 1 / p ∥ ( b j ) ∥ ℓ p CL(H_m)^{A+1/2-1/p}\|(b_j)\|_{\ell^p} C L ( H m ) A + 1/2 − 1/ p ∥ ( b j ) ∥ ℓ p
For the remaining j j j r < R j = H m / N j r<R_j=H_m/N_j r < R j = H m / N j D = ⌈ log 2 ( 2 H m ) ⌉ D=\lceil\log_2(2H_m)\rceil D = ⌈ log 2 ( 2 H m )⌉ R k + 1 < r ≤ R k R_{k+1}<r\leq R_k R k + 1 < r ≤ R k R k = H m 2 − k R_k=H_m2^{-k} R k = H m 2 − k k k k j ≤ k j\leq k j ≤ k k − D ≤ j ≤ k k-D\leq j\leq k k − D ≤ j ≤ k ( 1 + D ) 1 / 2 − 1 / p (1+D)^{1/2-1/p} ( 1 + D ) 1/2 − 1/ p p p p k k k v j v_j v j r ≤ R j r\leq R_j r ≤ R j C L ( H m ) A b j CL(H_m)^A b_j C L ( H m ) A b j j < k − D j<k-D j < k − D N j R k = H m 2 j − k ≤ 2 − ( k − j − D ) N_jR_k=H_m2^{j-k}\leq2^{-(k-j-D)} N j R k = H m 2 j − k ≤ 2 − ( k − j − D ) r ≤ R k r\leq R_k r ≤ R k C 2 − ( k − j − D ) n / p b j C2^{-(k-j-D)n/p}b_j C 2 − ( k − j − D ) n / p b j j j j ℓ p ( k ) \ell^p(k) ℓ p ( k ) m m m
C L ( H m ) A + 1 / 2 − 1 / p ∥ ( b j ) ∥ ℓ p ≲ 2 m ( A + 1 / 2 − 1 / p ) ∥ Q m f ∥ p . CL(H_m)^{A+1/2-1/p}\|(b_j)\|_{\ell^p}
\lesssim2^{m(A+1/2-1/p)}\|Q_mf\|_p. C L ( H m ) A + 1/2 − 1/ p ∥ ( b j ) ∥ ℓ p ≲ 2 m ( A + 1/2 − 1/ p ) ∥ Q m f ∥ p . In the last step we used p > 2 p>2 p > 2 Q m f Q_mf Q m f m m m C B ∥ L ( Λ ω ) B f ∥ p C_B\|L(\Lambda_\omega)^Bf\|_p C B ∥ L ( Λ ω ) B f ∥ p B > A + 1 / 2 − 1 / p B>A+1/2-1/p B > A + 1/2 − 1/ p
These estimates control the square function of the original u j u_j u j r r r u j u_j u j ( w j ) ↦ ∑ j Π N j w j (w_j) \mapsto\sum_j \Pi_{N_j}w_j ( w j ) ↦ ∑ j Π N j w j L q ( I × R n ; ℓ 2 ) L^q(I \times\mathbb{R}^n;\ell^2) L q ( I × R n ; ℓ 2 ) L q ( I × R n ) L^q(I \times\mathbb{R}^n) L q ( I × R n ) 1 < q < ∞ 1<q<\infty 1 < q < ∞ [32 Theorem 6.1.2 and Remark 6.1.3] . Real interpolation gives the same assertion for joint weak L p L^p L p L ( Λ ω ) − B L(\Lambda_\omega)^{-B} L ( Λ ω ) − B L ( Λ ω ) B Λ − ε L(\Lambda_\omega)^B\Lambda^{-\varepsilon} L ( Λ ω ) B Λ − ε ε > 0 \varepsilon>0 ε > 0 ∥ f ∥ p \|f\|_p ∥ f ∥ p
Disjoint packets and the weak spacetime norm
We prove Proposition 8.2. The geometry is the one used in Fefferman’s ball multiplier example [23 ] ; Wolff explains its relation to local smoothing in the discussion of (39) in [79 ] . For the quantitative compression factor ( log ( 1 / δ ) ) − ( n − 1 ) (\log(1/\delta))^{-(n-1)} ( log ( 1/ δ ) ) − ( n − 1 ) [24 Theorem 1.4] . We give the short construction needed here. The tubes are disjoint on one longitudinal interval. On a second interval, their union is small and the sum of their individual volumes stays bounded below. A modulated packet travels from the first part of its tube to the second.
An arrangement of tubes. Write x = ( X , Y ) ∈ R × R d x=(X,Y)\in\mathbb{R}\times\mathbb{R}^d x = ( X , Y ) ∈ R × R d d = n − 1 d=n-1 d = n − 1 k = ⌊ 1 4 log 2 N ⌋ k=\lfloor\frac{1}{4}\log_2 N\rfloor k = ⌊ 4 1 log 2 N ⌋ δ = 2 − k \delta=2^{-k} δ = 2 − k α = 1 / ( 4 d ) \alpha=1/(4\sqrt{d}) α = 1/ ( 4 d ) w = α δ / 16 w=\alpha\delta/16 w = α δ /16 0 ≤ j < 2 k 0\leq j<2^k 0 ≤ j < 2 k j δ = ∑ ℓ = 1 k ε ℓ ( j ) 2 − ℓ j\delta=\sum_{\ell=1}^{k}\varepsilon_\ell(j)2^{-\ell} j δ = ∑ ℓ = 1 k ε ℓ ( j ) 2 − ℓ s j = α j δ s_j=\alpha j\delta s j = α j δ
a ℓ = 1 − ℓ / k , L j ( X ) = α ∑ ℓ = 1 k ε ℓ ( j ) 2 − ℓ ( X − a ℓ ) . a_\ell=1-\ell/k,\qquad L_j(X)=\alpha\sum_{\ell=1}^{k}\varepsilon_\ell(j)2^{-\ell}(X-a_\ell). a ℓ = 1 − ℓ / k , L j ( X ) = α ℓ = 1 ∑ k ε ℓ ( j ) 2 − ℓ ( X − a ℓ ) . The ℓ \ell ℓ X = a ℓ X=a_\ell X = a ℓ X ≤ − 1 / 2 X\leq-1/2 X ≤ − 1/2 j j j j + 1 j+1 j + 1 m m m
L j + 1 ( X ) − L j ( X ) = α δ ( X − a m ) + α ∑ ℓ > m 2 − ℓ ( a ℓ − a m ) ≤ − α δ / 2. L_{j+1}(X)-L_j(X)=\alpha\delta(X-a_m)+\alpha\sum_{\ell>m}2^{-\ell}(a_\ell-a_m)\leq-\alpha\delta/2. L j + 1 ( X ) − L j ( X ) = α δ ( X − a m ) + α ℓ > m ∑ 2 − ℓ ( a ℓ − a m ) ≤ − α δ /2. The intervals of radius w w w L j ( X ) L_j(X) L j ( X )
For X X X J = [ 1 / 4 , 1 / 2 ] J=[1/4,1/2] J = [ 1/4 , 1/2 ] B X B_X B X w / 2 w/2 w /2 L j ( X ) L_j(X) L j ( X ) m = ⌊ k ( 1 − X ) ⌋ m=\lfloor k(1-X)\rfloor m = ⌊ k ( 1 − X )⌋ ℓ > m \ell>m ℓ > m 0 ≤ X − a ℓ ≤ ( ℓ − m ) / k 0\leq X-a_\ell\leq(\ell-m)/k 0 ≤ X − a ℓ ≤ ( ℓ − m ) / k α ∑ ℓ > m 2 − ℓ ( ℓ − m ) / k ≤ 2 α 2 − m / k \alpha\sum_{\ell>m}2^{-\ell}(\ell-m)/k\leq2\alpha2^{-m}/k α ∑ ℓ > m 2 − ℓ ( ℓ − m ) / k ≤ 2 α 2 − m / k 2 m 2^m 2 m m ≤ 3 k / 4 m\leq3k/4 m ≤ 3 k /4
∣ B X ∣ ≤ 2 α / k + 2 m w ≲ α / k . |B_X|\leq2\alpha/k+2^m w\lesssim\alpha/k. ∣ B X ∣ ≤ 2 α / k + 2 m w ≲ α / k . Taking an independent choice in each transverse coordinate produces δ − d \delta^{-d} δ − d ν = ( j 1 , … , j d ) \nu=(j_1,\ldots,j_d) ν = ( j 1 , … , j d ) L ν = ( L j 1 , … , L j d ) L_\nu=(L_{j_1},\ldots,L_{j_d}) L ν = ( L j 1 , … , L j d ) s ν = ( s j 1 , … , s j d ) s_\nu=(s_{j_1},\ldots,s_{j_d}) s ν = ( s j 1 , … , s j d )
S ν = { ( X , Y ) : X ∈ J , ∣ Y i − L j i ( X ) ∣ ≤ w / 2 , 1 ≤ i ≤ d } S_\nu=\{(X,Y):X\in J,\ |Y_i-L_{j_i}(X)|\leq w/2,\ 1\leq i\leq d\} S ν = {( X , Y ) : X ∈ J , ∣ Y i − L j i ( X ) ∣ ≤ w /2 , 1 ≤ i ≤ d } all lie in E = { ( X , Y ) : X ∈ J , Y ∈ ( B X ) d } E=\{(X,Y):X\in J,\ Y\in(B_X)^d\} E = {( X , Y ) : X ∈ J , Y ∈ ( B X ) d }
∣ E ∣ ≲ n k − d , ∑ ν ∣ S ν ∣ = δ − d ∣ J ∣ w d ∼ n 1. (95) |E|\lesssim_n k^{-d},\qquad\sum_\nu|S_\nu|=\delta^{-d}|J|w^d\sim_n 1. \tag*{(95)} ∣ E ∣ ≲ n k − d , ν ∑ ∣ S ν ∣ = δ − d ∣ J ∣ w d ∼ n 1. (95) Transport at frequency N N N Choose 0 ≤ χ , θ ≤ 1 0 \le\chi,\theta\le1 0 ≤ χ , θ ≤ 1 χ \chi χ ( − 2 , − 1 / 2 ) (-2,-1/2) ( − 2 , − 1/2 ) [ − 3 / 2 , − 3 / 4 ] [-3/2,-3/4] [ − 3/2 , − 3/4 ] θ \theta θ ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) [ − 1 / 2 , 1 / 2 ] [-1/2,1/2] [ − 1/2 , 1/2 ]
b ν ( X , Y ) = χ ( X ) ∏ i = 1 d θ ( ( Y i − L j i ( X ) ) / w ) , v ν = ( 1 , s ν ) ( 1 + ∣ s ν ∣ 2 ) − 1 / 2 . b_\nu(X,Y)=\chi(X)\prod_{i=1}^{d}\theta\left((Y_i-L_{j_i}(X))/w\right),\qquad v_\nu=(1,s_\nu)(1+|s_\nu|^2)^{-1/2}. b ν ( X , Y ) = χ ( X ) i = 1 ∏ d θ ( ( Y i − L j i ( X )) / w ) , v ν = ( 1 , s ν ) ( 1 + ∣ s ν ∣ 2 ) − 1/2 . The b ν b_\nu b ν O n ( δ − d w d ) = O n ( 1 ) O_n(\delta^{-d}w^d)=O_n(1) O n ( δ − d w d ) = O n ( 1 ) λ = 3 N / 2 \lambda=3N/2 λ = 3 N /2 f ν ( x ) = e i λ v ν ⋅ x b ν ( x ) f_\nu(x)=e^{i\lambda v_\nu\cdot x}b_\nu(x) f ν ( x ) = e iλ v ν ⋅ x b ν ( x ) Q N = β ( ∣ D ∣ / N ) Q_N=\beta(|D|/N) Q N = β ( ∣ D ∣/ N ) β ∈ C ∞ ( ( 5 / 4 , 7 / 4 ) ) \beta\in C^\infty((5/4,7/4)) β ∈ C ∞ (( 5/4 , 7/4 )) 3 / 2 3/2 3/2 ν , x \nu,x ν , x 1 ≤ t ≤ 2 1\le t\le2 1 ≤ t ≤ 2
U ( t ) Q N f ν ( x ) = e i λ ( v ν ⋅ x − t ) b ν ( x − t v ν ) + O n ( N − 1 w − 2 ) . (96) U(t)Q_Nf_\nu(x)=e^{i\lambda(v_\nu\cdot x-t)}b_\nu(x-tv_\nu)+O_n(N^{-1}w^{-2}). \tag*{(96)} U ( t ) Q N f ν ( x ) = e iλ ( v ν ⋅ x − t ) b ν ( x − t v ν ) + O n ( N − 1 w − 2 ) . (96) To verify this, write ξ = λ v ν + η \xi=\lambda v_\nu+\eta ξ = λ v ν + η ∣ η ∣ ≤ c N |\eta|\le cN ∣ η ∣ ≤ c N ∣ ξ ∣ = λ + v ν ⋅ η + O ( ∣ η ∣ 2 / N ) |\xi|=\lambda+v_\nu\cdot\eta+O(|\eta|^2/N) ∣ ξ ∣ = λ + v ν ⋅ η + O ( ∣ η ∣ 2 / N )
w d ∣ χ ^ ( η 1 + s ν ⋅ η ′ ) ∣ ∏ i = 1 d ∣ θ ^ ( w η i ′ ) ∣ , η = ( η 1 , η ′ ) . w^d\left|\widehat{\chi}(\eta_1+s_\nu\cdot\eta')\right|\prod_{i=1}^{d}\left|\widehat{\theta}(w\eta_i')\right|,\qquad\eta=(\eta_1,\eta'). w d ∣ χ ( η 1 + s ν ⋅ η ′ ) ∣ i = 1 ∏ d θ ( w η i ′ ) , η = ( η 1 , η ′ ) . The change of variables in these arguments and ∣ s ν ∣ ≤ 1 / 4 |s_\nu|\le1/4 ∣ s ν ∣ ≤ 1/4 ∫ ∣ η ∣ 2 ∣ b ^ ν ( η ) ∣ d η ≲ n w − 2 \int|\eta|^2|\widehat b_\nu(\eta)|\,d\eta\lesssim_n w^{-2} ∫ ∣ η ∣ 2 ∣ b ν ( η ) ∣ d η ≲ n w − 2 O n ( N − 1 w − 2 ) O_n(N^{-1}w^{-2}) O n ( N − 1 w − 2 ) ∣ η ∣ > c N |\eta|>cN ∣ η ∣ > c N O n ( N − 2 w − 2 ) O_n(N^{-2}w^{-2}) O n ( N − 2 w − 2 ) w ≲ n δ w\lesssim_n\delta w ≲ n δ N ≥ δ − 4 N\ge\delta^{-4} N ≥ δ − 4
Take I 0 = [ 3 / 2 , 7 / 4 ] I_0=[3/2,7/4] I 0 = [ 3/2 , 7/4 ] t v ν tv_\nu t v ν Y i − L j i ( X ) Y_i-L_{j_i}(X) Y i − L j i ( X ) X ∈ J X\in J X ∈ J t ∈ I 0 t\in I_0 t ∈ I 0 − 3 / 2 ≤ X − t ( v ν ) 1 ≤ − 3 / 4 -3/2\le X-t(v_\nu)_1\le-3/4 − 3/2 ≤ X − t ( v ν ) 1 ≤ − 3/4 4 / 17 ≤ ( v ν ) 1 ≤ 1 4/\sqrt{17}\le(v_\nu)_1\le1 4/ 17 ≤ ( v ν ) 1 ≤ 1 b ν ( x − t v ν ) = 1 b_\nu(x-tv_\nu)=1 b ν ( x − t v ν ) = 1 S ν × I 0 S_\nu\times I_0 S ν × I 0 ∣ U ( t ) Q N f ν ∣ ≥ 1 / 2 |U(t)Q_Nf_\nu|\ge1/2 ∣ U ( t ) Q N f ν ∣ ≥ 1/2 N N N
Choosing the signs. For independent random signs ϵ ν \epsilon_\nu ϵ ν f = ∑ ν ϵ ν f ν f=\sum_\nu\epsilon_\nu f_\nu f = ∑ ν ϵ ν f ν ∥ f ∥ p ≲ n , p 1 \|f\|_p\lesssim_{n,p}1 ∥ f ∥ p ≲ n , p 1
E ∫ E × I 0 ∣ U ( t ) Q N f ( x ) ∣ 2 d x d t = ∑ ν ∫ E × I 0 ∣ U ( t ) Q N f ν ( x ) ∣ 2 d x d t ≥ 1 4 ∣ I 0 ∣ ∑ ν ∣ S ν ∣ ≳ n 1. \mathbb{E}\int_{E\times I_0}|U(t)Q_Nf(x)|^2\,dx\,dt
=\sum_\nu\int_{E\times I_0}|U(t)Q_Nf_\nu(x)|^2\,dx\,dt
\ge\frac14|I_0|\sum_\nu|S_\nu|\gtrsim_n1. E ∫ E × I 0 ∣ U ( t ) Q N f ( x ) ∣ 2 d x d t = ν ∑ ∫ E × I 0 ∣ U ( t ) Q N f ν ( x ) ∣ 2 d x d t ≥ 4 1 ∣ I 0 ∣ ν ∑ ∣ S ν ∣ ≳ n 1. Fix signs attaining this lower bound and set g N = Q N f g_N=Q_Nf g N = Q N f ∥ g N ∥ p ≲ n , p 1 \|g_N\|_p\lesssim_{n,p}1 ∥ g N ∥ p ≲ n , p 1 L 1 L^1 L 1 Q N Q_N Q N p > 2 p>2 p > 2 H H H
∫ H ∣ u ∣ 2 ≤ p p − 2 ∣ H ∣ 1 − 2 / p ∥ u ∥ L p , ∞ 2 . \int_H|u|^2\le\frac{p}{p-2}|H|^{1-2/p}\|u\|_{L^{p,\infty}}^2. ∫ H ∣ u ∣ 2 ≤ p − 2 p ∣ H ∣ 1 − 2/ p ∥ u ∥ L p , ∞ 2 . Taking H = E × I 0 H=E\times I_0 H = E × I 0 ∣ E ∣ ≲ n k − d |E|\lesssim_n k^{-d} ∣ E ∣ ≲ n k − d ∥ U ( t ) g N ∥ L p , ∞ ≳ n , p k d ( 1 / 2 − 1 / p ) \|U(t)g_N\|_{L^{p,\infty}}\gtrsim_{n,p}k^{d(1/2-1/p)} ∥ U ( t ) g N ∥ L p , ∞ ≳ n , p k d ( 1/2 − 1/ p ) k ≍ log N k\asymp\log N k ≍ log N Proposition 8.2 . In fact the inputs have bounded L 2 L^2 L 2 L ∞ L^\infty L ∞ L p , 1 L^{p,1} L p , 1
These packets also test the angular weight in Theorem B.1 . At p = p c p=p_c p = p c [ log ( 2 + Λ ω ) ] B [\log(2+\Lambda_\omega)]^B [ log ( 2 + Λ ω ) ] B B ≥ 1 / p B\ge1/p B ≥ 1/ p C N CN C N C N 2 CN^2 C N 2 w − 1 ≲ n N 1 / 4 w^{-1}\lesssim_nN^{1/4} w − 1 ≲ n N 1/4 x i ∂ j − x j ∂ i x_i\partial_j-x_j\partial_i x i ∂ j − x j ∂ i ∥ ( 1 + Λ ω 2 / N 2 ) g N ∥ p ≲ 1 \|(1+\Lambda_\omega^2/N^2)g_N\|_p\lesssim1 ∥ ( 1 + Λ ω 2 / N 2 ) g N ∥ p ≲ 1 B ≥ 0 B\ge0 B ≥ 0 [ log ( 2 + Λ ω ) ] B / ( 1 + Λ ω 2 / N 2 ) [\log(2+\Lambda_\omega)]^B/(1+\Lambda_\omega^2/N^2) [ log ( 2 + Λ ω ) ] B / ( 1 + Λ ω 2 / N 2 ) O B ( ( log ( 2 + N ) ) B ) O_B((\log(2+N))^B) O B (( log ( 2 + N ) ) B ) B < 0 B<0 B < 0 B ≥ 1 / p B\ge1/p B ≥ 1/ p Theorem B.1 gives sufficiency for B > 1 / 2 + 1 / p B>1/2+1/p B > 1/2 + 1/ p
A linear Dirichlet problem for the planar wave
When n = 2 n=2 n = 2 p c = 4 p_c=4 p c = 4 Theorem 1.2 with a linear auxiliary Dirichlet problem on a disk. We give the reduction, including the boundary estimate needed to apply Lemma 6.1 .
The disk equation. Fix nonzero periodic data in a single cap as in Proposition 5.1 . Retain z = ( T , τ ) z=(T,\tau) z = ( T , τ ) s = ∣ τ ∣ s=|\tau| s = ∣ τ ∣ a = K h a=Kh a = K h Q = Φ / a Q=\Phi/a Q = Φ/ a a < s < 4 a<s<4 a < s < 4
Ω z p l : r W = ∣ W ∣ 2 w 0 2 − 1 < 0 , ∣ Q ∣ < 1 10 . \Omega_z^{\mathrm{pl}}:\qquad r_W=\frac{|W|^2}{w_0^2}-1<0,\qquad|Q|<\frac{1}{10}. Ω z pl : r W = w 0 2 ∣ W ∣ 2 − 1 < 0 , ∣ Q ∣ < 10 1 . The real variable X X X τ \tau τ (33) gives a holomorphic chart from this product to the imaginary spatial variables ( P , Z ) (P,Z) ( P , Z ) A = A w = g ( Q ) ( J h ∗ X A 0 ) A=A_w=g(Q)(J_h*_{X}A^0) A = A w = g ( Q ) ( J h ∗ X A 0 ) (36) , with g = 1 g=1 g = 1 ∣ Q ∣ ≤ 1 / 100 |Q|\leq1/100 ∣ Q ∣ ≤ 1/100 g = 0 g=0 g = 0 ∣ Q ∣ ≥ 1 / 20 |Q|\geq1/20 ∣ Q ∣ ≥ 1/20 μ = A T T \mu=A_{TT} μ = A T T
Put B = H ( ∣ Φ ∣ 2 ) B=H(|\Phi|^2) B = H ( ∣Φ ∣ 2 ) B f i b B_{\mathrm{fib}} B fib L B u = C ( B f i b ) : H f i b ( u ) L_Bu=C(B_{\mathrm{fib}}):H_{\mathrm{fib}}(u) L B u = C ( B fib ) : H fib ( u ) c = ( ∂ P Φ , ∂ Z Φ ) c=(\partial_P\Phi,\partial_Z\Phi) c = ( ∂ P Φ , ∂ Z Φ ) e = ( c 2 , − c 1 ) t e=(c_2,-c_1)^{\mathsf t} e = ( c 2 , − c 1 ) t B f i b = 4 c ∗ c B_{\mathrm{fib}}=4c^*c B fib = 4 c ∗ c C ( B f i b ) = 4 e e ∗ C(B_{\mathrm{fib}})=4ee^* C ( B fib ) = 4 e e ∗
J = det C ∂ ( W , Φ ) ∂ ( P , Z ) = 1 + W 2 / 2 2 Y . \mathcal{J}=\det_{\mathbb{C}}\frac{\partial(W,\Phi)}{\partial(P,Z)}=\frac{1+W^2/2}{\sqrt{2}Y}. J = C det ∂ ( P , Z ) ∂ ( W , Φ ) = 2 Y 1 + W 2 /2 . Then
L B u = 4 ∑ i , j = 1 2 Re ( e i e j ‾ ) ∂ X i X j u + 4 ∣ J ∣ 2 Δ W u , ∣ J ∣ ≍ s − 1 . (97) L_Bu=4\sum_{i,j=1}^{2}\operatorname{Re}(e_i\overline{e_j})\partial_{X_iX_j}u+4|\mathcal{J}|^2\Delta_Wu,\qquad|\mathcal{J}|\asymp s^{-1}. \tag*{(97)} L B u = 4 i , j = 1 ∑ 2 Re ( e i e j ) ∂ X i X j u + 4∣ J ∣ 2 Δ W u , ∣ J ∣ ≍ s − 1 . (97) Indeed, the mixed derivatives between X X X ( P , Z ) (P,Z) ( P , Z ) C k \mathbb{C}_k C k e e ∗ ee^* e e ∗ C k \mathbb{C}_k C k Lemma 3.1 ; under the holomorphic chart it pairs with the vector ( ∂ W e , c e ) = ( J , 0 ) (\partial_W e,ce)=(\mathcal{J},0) ( ∂ W e , ce ) = ( J , 0 ) (33) .
Solve L B u = μ L_Bu=\mu L B u = μ u = 0 u=0 u = 0 ∣ W ∣ = w 0 |W|=w_0 ∣ W ∣ = w 0 ( z , Φ ) (z,\Phi) ( z , Φ ) k k k
4 ∣ J ∣ 2 Δ W u ^ − 4 ∣ k ⋅ e ∣ 2 u ^ = μ ^ ( ∣ W ∣ < w 0 ) , u ^ ∣ ∣ W ∣ = w 0 = 0. 4|\mathcal{J}|^2\Delta_W\widehat{u}-4|k\cdot e|^2\widehat{u}=\widehat{\mu}\qquad(|W|<w_0),\qquad\widehat{u}\big|_{|W|=w_0}=0. 4∣ J ∣ 2 Δ W u − 4∣ k ⋅ e ∣ 2 u = μ ( ∣ W ∣ < w 0 ) , u ∣ W ∣ = w 0 = 0. Here Δ W \Delta_W Δ W 4 ∣ J ∣ 2 4|\mathcal{J}|^2 4∣ J ∣ 2 k = 0 k=0 k = 0 μ \mu μ g ( Q ) g(Q) g ( Q ) u = g ( Q ) u ~ u=g(Q)\widetilde{u} u = g ( Q ) u u ~ \widetilde{u} u u u u Q Q Q u ≤ 0 u\leq0 u ≤ 0 ϵ > 0 \epsilon>0 ϵ > 0 L B ( u + ϵ r W ) = μ + 16 ϵ ∣ J ∣ 2 / w 0 2 > 0 L_B(u+\epsilon r_W)=\mu+16\epsilon|\mathcal{J}|^2/w_0^2>0 L B ( u + ϵ r W ) = μ + 16 ϵ ∣ J ∣ 2 / w 0 2 > 0 ( X , W ) (X,W) ( X , W ) ϵ ↓ 0 \epsilon\downarrow0 ϵ ↓ 0
The energy. For the full three-by-three Hessian H = H ( u ) H=H(u) H = H ( u ) D B ( H ) = B : C ( H ) \mathcal{D}_B(H)=B:C(H) D B ( H ) = B : C ( H ) λ \lambda λ det ( λ B + H ) \det(\lambda B+H) det ( λ B + H ) H H H M M M H H H Lemma 3.3 . Homogeneity gives M : H = 2 D B ( H ) M:H=2\mathcal{D}_B(H) M : H = 2 D B ( H ) T T TT T T M T T = C ( B f i b ) : H f i b ( u ) = μ M_{TT}=C(B_{\mathrm{fib}}):H_{\mathrm{fib}}(u)=\mu M T T = C ( B fib ) : H fib ( u ) = μ W W W H ( r W ) H(r_W) H ( r W ) ν = G ( r W ) / ∣ ∇ r W ∣ \nu=G(r_W)/|\nabla r_W| ν = G ( r W ) /∣∇ r W ∣ u = 0 u=0 u = 0 u = ( u / r W ) r W u=(u/r_W)r_W u = ( u / r W ) r W ν ⊥ \nu^\perp ν ⊥ H ∣ ν ⊥ = 0 H|_{\nu^\perp}=0 H ∣ ν ⊥ = 0 λ \lambda λ det ( ( λ B + H ) ∣ ν ⊥ ) \det((\lambda B+H)|_{\nu^\perp}) det (( λ B + H ) ∣ ν ⊥ ) ν ∗ M ν = 0 \nu^*M\nu=0 ν ∗ M ν = 0 Lemma 3.2 . Now set
X 0 ( z ) = ∫ Ω z p l ( − u ) μ d ζ , F p l ( z ) = 1 2 X 0 ( z ) , d ζ = d X d P d Z . X_0(z)=\int_{\Omega_z^{\mathrm{pl}}}(-u)\mu\,d\zeta,\qquad\mathcal{F}_{\mathrm{pl}}(z)=\frac{1}{2}X_0(z),\qquad d\zeta=dX\,dP\,dZ. X 0 ( z ) = ∫ Ω z pl ( − u ) μ d ζ , F pl ( z ) = 2 1 X 0 ( z ) , d ζ = d X d P d Z . Green’s formula (44) for M M M Δ z X 0 = − 2 ∫ D B ( H ) d ζ \Delta_z X_0=-2\int\mathcal{D}_B(H)\,d\zeta Δ z X 0 = − 2 ∫ D B ( H ) d ζ Q Q Q u u u
To apply the same formula to A A A A : H ≥ D B ( H ) A:H\geq\mathcal{D}_B(H) A : H ≥ D B ( H ) E = ker B E=\ker B E = ker B β = tr B \beta=\operatorname{tr}B β = tr B β f = tr B f i b \beta_f=\operatorname{tr}B_{\mathrm{fib}} β f = tr B fib E E E e T e_T e T ( 0 , e / ∣ e ∣ ) (0,e/|e|) ( 0 , e /∣ e ∣ ) α = A ∣ E \alpha=A|_E α = A ∣ E b = H ∣ E b=H|_E b = H ∣ E e T e_T e T β f / β \beta_f/\beta β f / β α 00 β f / β = A T T = μ = β f b 11 \alpha_{00}\beta_f/\beta=A_{TT}=\mu=\beta_f b_{11} α 00 β f / β = A T T = μ = β f b 11 D B ( H ) = β det b \mathcal{D}_B(H)=\beta\det b D B ( H ) = β det b α 00 = β b 11 \alpha_{00}=\beta b_{11} α 00 = β b 11
A : H − D B ( H ) = β ∣ b 01 + α 01 / β ∣ 2 + det ( α ) / β ≥ 0. A:H-\mathcal{D}_B(H)=\beta\left|b_{01}+\alpha_{01}/\beta\right|^2+\det(\alpha)/\beta\geq0. A : H − D B ( H ) = β ∣ b 01 + α 01 / β ∣ 2 + det ( α ) / β ≥ 0. Green’s formula for A A A Δ z X 0 \Delta_zX_0 Δ z X 0 1 2 Δ z X 0 \frac{1}{2}\Delta_zX_0 2 1 Δ z X 0
F p l ≥ 0 , Δ z F p l ≤ ∫ ∣ Q ∣ < 1 / 10 δ ( r W ) ( u / r W ) G ( r W ) ∗ A G ( r W ) d ζ . (98) \mathcal{F}_{\mathrm{pl}}\geq0,\qquad
\Delta_z\mathcal{F}_{\mathrm{pl}}\leq\int_{|Q|<1/10}\delta(r_W)(u/r_W)G(r_W)^*AG(r_W)\,d\zeta. \tag*{(98)} F pl ≥ 0 , Δ z F pl ≤ ∫ ∣ Q ∣ < 1/10 δ ( r W ) ( u / r W ) G ( r W ) ∗ A G ( r W ) d ζ . (98) Comparison on the product fiber. Write σ = s / h \sigma=s/h σ = s / h g T = e − i T ∣ D ∣ f g_T=e^{-iT|D|}f g T = e − i T ∣ D ∣ f P ∗ = J D s ∗ ∣ g T ∣ 2 P_*=J_{D_s}*|g_T|^2 P ∗ = J D s ∗ ∣ g T ∣ 2 D D D (50) applies throughout the product: y ≍ s y\asymp s y ≍ s ∣ W ∣ = w 0 |W|=w_0 ∣ W ∣ = w 0 W W W ∥ A ∥ ≲ s − 2 σ m P ∗ \|A\|\lesssim s^{-2}\sigma^mP_* ∥ A ∥ ≲ s − 2 σ m P ∗ e − c 0 σ e^{-c_0\sigma} e − c 0 σ (49) ,
L B ( r W P ∗ ) = 16 ∣ J ∣ 2 P ∗ / w 0 2 + r W L B P ∗ ≥ ( c − C D − 2 ) s − 2 P ∗ . L_B(r_WP_*)=16|\mathcal{J}|^2P_*/w_0^2+r_WL_BP_*\geq(c-CD^{-2})s^{-2}P_*. L B ( r W P ∗ ) = 16∣ J ∣ 2 P ∗ / w 0 2 + r W L B P ∗ ≥ ( c − C D − 2 ) s − 2 P ∗ . The two variables separate, so no mixed gradient term occurs. For fixed large C 1 , m 1 C_1,m_1 C 1 , m 1 L B ( C 1 σ m 1 r W P ∗ ) > μ L_B(C_1\sigma^{m_1}r_WP_*)>\mu L B ( C 1 σ m 1 r W P ∗ ) > μ u u u W W W Q Q Q 0 ≤ u / r W ≤ C 1 σ m 1 P ∗ 0\leq u/r_W\leq C_1\sigma^{m_1}P_* 0 ≤ u / r W ≤ C 1 σ m 1 P ∗ d P d Z = a 2 ∣ J ∣ − 2 d W d Q dP\,dZ=a^2|\mathcal{J}|^{-2}\,dW\,dQ d P d Z = a 2 ∣ J ∣ − 2 d W d Q ∫ δ ( r W ) d W = π w 0 2 \int\delta(r_W)\,dW=\pi w_0^2 ∫ δ ( r W ) d W = π w 0 2 ∣ G ( r W ) ∣ ≲ s − 1 |G(r_W)|\lesssim s^{-1} ∣ G ( r W ) ∣ ≲ s − 1 ∫ P ∗ 2 d X ≤ ∥ g T ∥ 4 4 \int P_*^2\,dX\leq\|g_T\|_4^4 ∫ P ∗ 2 d X ≤ ∥ g T ∥ 4 4
F p l ≲ a 2 σ m ∥ g T ∥ 4 4 , Δ z F p l ≲ ( a / s ) 2 σ m e − c 0 σ ∥ g T ∥ 4 4 . \mathcal{F}_{\mathrm{pl}}\lesssim a^2\sigma^m\|g_T\|_4^4,\qquad
\Delta_z\mathcal{F}_{\mathrm{pl}}\lesssim(a/s)^2\sigma^m e^{-c_0\sigma}\|g_T\|_4^4. F pl ≲ a 2 σ m ∥ g T ∥ 4 4 , Δ z F pl ≲ ( a / s ) 2 σ m e − c 0 σ ∥ g T ∥ 4 4 . For the lower bound use the centers ∣ W ∣ ≤ γ |W|\leq\gamma ∣ W ∣ ≤ γ ∣ Q ∣ ≤ 1 / 400 |Q|\leq1/400 ∣ Q ∣ ≤ 1/400 § 5 . The inverse multiplier in the proof of (56) bounds the wave density itself below by m X m_X m X h h h d = 1 d=1 d = 1 l l l H f i b ( l 2 ) = 6 I H_{\mathrm{fib}}(l^2)=6I H fib ( l 2 ) = 6 I L B ( l 2 ) = 24 ∣ e ∣ 2 ≤ C L_B(l^2)=24|e|^2\leq C L B ( l 2 ) = 24∣ e ∣ 2 ≤ C m X > 0 m_X>0 m X > 0 ϵ m X ( l 2 − h 2 ) \epsilon m_X(l^2-h^2) ϵ m X ( l 2 − h 2 ) u u u ϵ \epsilon ϵ L B L_B L B ( − u ) μ ≥ ϵ h 2 m X 2 (-u)\mu\geq\epsilon h^2m_X^2 ( − u ) μ ≥ ϵ h 2 m X 2 m X = 0 m_X=0 m X = 0
F p l ≳ γ a 2 σ − m e − C γ 2 σ ∥ g T ∥ 4 4 . \mathcal{F}_{\mathrm{pl}}\gtrsim_\gamma a^2\sigma^{-m}e^{-C\gamma^2\sigma}\|g_T\|_4^4. F pl ≳ γ a 2 σ − m e − C γ 2 σ ∥ g T ∥ 4 4 . since the imaginary center region has volume comparable to a 2 s 2 γ 2 a^2s^2\gamma^2 a 2 s 2 γ 2 h 2 s 2 ( h s ) − 2 = 1 h^2s^2(hs)^{-2}=1 h 2 s 2 ( h s ) − 2 = 1 γ \gamma γ K K K Proposition 5.1 for F p l \mathcal{F}_{\mathrm{pl}} F pl Lemma 6.1 and § 7 now give the planar O ( ( log ( 2 / h ) ) 1 / 2 ) O((\log(2/h))^{1/2}) O (( log ( 2/ h ) ) 1/2 )
Curved characteristic sheets of symmetric systems
In the proof of Theorem 1.2 , the wave symbol supplies both a polarization for each plane wave and a Gaussian centered at that polarization. A curved sheet of the largest eigenvalue of a symmetric matrix pencil supplies the same two objects. We construct them on one angular patch and check the two places where the phase enters the fiber estimate: Gaussian decay on the wall and recovery of the wave near the center.
Theorem E.1. Let C 1 , … , C n C_1,\ldots,C_n C 1 , … , C n n × n n \times n n × n n ≥ 2 n \ge2 n ≥ 2 C ( ξ ) = ∑ j = 1 n ξ j C j C(\xi)=\sum_{j=1}^{n}\xi_jC_j C ( ξ ) = ∑ j = 1 n ξ j C j ϕ ( ξ ) = λ max ( C ( ξ ) ) \phi(\xi)=\lambda_{\max}(C(\xi)) ϕ ( ξ ) = λ m a x ( C ( ξ )) Γ ⊂ R n ∖ { 0 } \Gamma\subset\mathbb{R}^n \setminus\{0\} Γ ⊂ R n ∖ { 0 } D 2 ϕ ( ξ ) D^2\phi(\xi) D 2 ϕ ( ξ ) ξ ⊥ \xi^\perp ξ ⊥ b ∈ C c ∞ ( Γ ) b \in C^\infty_c(\Gamma) b ∈ C c ∞ ( Γ ) 0 < h ≤ 1 0<h\le1 0 < h ≤ 1 f f f R n \mathbb{R}^n R n
∥ e − i T ϕ ( D ) b ( h D ) f ∥ L p c ( R n × [ 1 , 2 ] ) ≤ C b [ log ( 2 / h ) ] 2 / p c ∥ f ∥ L p c ( R n ) , p c = 2 n n − 1 . (99) \left\|e^{-iT\phi(D)}b(hD)f\right\|_{L^{p_c}(\mathbb{R}^n\times[1,2])}\le C_b[\log(2/h)]^{2/p_c}\|f\|_{L^{p_c}(\mathbb{R}^n)},\qquad p_c=\frac{2n}{n-1}. \tag*{(99)} e − i T ϕ ( D ) b ( h D ) f L p c ( R n × [ 1 , 2 ]) ≤ C b [ log ( 2/ h ) ] 2/ p c ∥ f ∥ L p c ( R n ) , p c = n − 1 2 n . (99) The constant may depend on the pencil. The same assertion holds for a simple smallest eigenvalue with negative definite transverse Hessian.
Example E.2 (A cubic characteristic sheet). In three spatial dimensions take the pencil
C ( x , y , z ) = ( z x y x − z 0 y 0 − 2 z ) . C(x,y,z)=\begin{pmatrix} z & x & y \\ x & -z & 0 \\ y & 0 & -2z \end{pmatrix}. C ( x , y , z ) = z x y x − z 0 y 0 − 2 z . At ( 0 , 0 , 1 ) (0,0,1) ( 0 , 0 , 1 ) 1 1 1 diag ( 1 , 2 / 3 ) \operatorname{diag}(1,2/3) diag ( 1 , 2/3 ) L 3 L^3 L 3
( t − z ) ( t + z ) ( t + 2 z ) − x 2 ( t + 2 z ) − y 2 ( t + z ) . (t-z)(t+z)(t+2z)-x^2(t+2z)-y^2(t+z). ( t − z ) ( t + z ) ( t + 2 z ) − x 2 ( t + 2 z ) − y 2 ( t + z ) . If this homogeneous cubic factored over C \mathbb{C} C t t t t = α x + β y + δ z t=\alpha x+\beta y+\delta z t = α x + β y + δ z z = 0 z=0 z = 0 t ( t 2 − x 2 − y 2 ) t(t^2-x^2-y^2) t ( t 2 − x 2 − y 2 ) x 2 + y 2 x^2+y^2 x 2 + y 2 α = β = 0 \alpha=\beta=0 α = β = 0 x 2 z x^2z x 2 z y 2 z y^2z y 2 z δ = − 2 \delta=-2 δ = − 2 δ = − 1 \delta=-1 δ = − 1 t t t ( x , y , z ) (x,y,z) ( x , y , z )
Eigenvectors and the spatial fiber
Fix a unit direction ω ∈ Γ \omega\in\Gamma ω ∈ Γ C ( ω ) C(\omega) C ( ω ) e 0 e_0 e 0 d = n − 1 d=n-1 d = n − 1 ω \omega ω [43 Chapter II] . Applying it to e 0 e_0 e 0 E ( ξ ) = ( 1 , v ( ξ ) ) t E(\xi)=(1,v(\xi))^t E ( ξ ) = ( 1 , v ( ξ ) ) t S ( ξ ) = ϕ ( ξ ) I − C ( ξ ) S(\xi)=\phi(\xi)I-C(\xi) S ( ξ ) = ϕ ( ξ ) I − C ( ξ ) d × d d\times d d × d S ⊥ ( ξ ) S_\perp(\xi) S ⊥ ( ξ )
S ( ξ ) E ( ξ ) = 0 , c ∣ ξ ∣ I ≤ S ⊥ ( ξ ) ≤ C ∣ ξ ∣ I . S(\xi)E(\xi)=0,\qquad c|\xi|I\le S_\perp(\xi)\le C|\xi|I. S ( ξ ) E ( ξ ) = 0 , c ∣ ξ ∣ I ≤ S ⊥ ( ξ ) ≤ C ∣ ξ ∣ I . Here and below constants are uniform on each chosen compact angular chart. Write L w Lw L w C ( w ) e 0 C(w)e_0 C ( w ) e 0 e 0 ⊥ e_0^\perp e 0 ⊥ ω + σ w \omega+\sigma w ω + σ w σ = 0 \sigma=0 σ = 0 S ⊥ ( ω ) − 1 L w S_\perp(\omega)^{-1}Lw S ⊥ ( ω ) − 1 L w e 0 ⊥ e_0^\perp e 0 ⊥ ϕ w ′ = e t C ( w ) e \phi'_w=e^tC(w)e ϕ w ′ = e t C ( w ) e
D 2 ϕ ( ω ) [ w , w ] = 2 ( L w ) t S ⊥ ( ω ) − 1 L w . (100) D^2\phi(\omega)[w,w]=2(Lw)^tS_\perp(\omega)^{-1}Lw. \tag*{(100)} D 2 ϕ ( ω ) [ w , w ] = 2 ( L w ) t S ⊥ ( ω ) − 1 L w . (100) Consequently L : ω ⊥ → e 0 ⊥ L:\omega^\perp\to e_0^\perp L : ω ⊥ → e 0 ⊥ D v ( ω ) = S ⊥ ( ω ) − 1 L Dv(\omega)=S_\perp(\omega)^{-1}L D v ( ω ) = S ⊥ ( ω ) − 1 L ω ⊥ \omega^\perp ω ⊥ v v v W ∈ C d W\in\mathbb{C}^d W ∈ C d e ( W ) = ( 1 , W ) t e(W)=(1,W)^t e ( W ) = ( 1 , W ) t
B j ( W ) = e ( W ) t C j e ( W ) 1 + W ⋅ W . B_j(W)=\frac{e(W)^tC_je(W)}{1+W\cdot W}. B j ( W ) = 1 + W ⋅ W e ( W ) t C j e ( W ) . e n d s m i d = 1 {ends_mid=1} e n d s m i d = 1 The products here are bilinear. On real eigenvectors this map is the group velocity: B ( v ( ξ ) ) = ∇ ϕ ( ξ ) B(v(\xi))=\nabla\phi(\xi) B ( v ( ξ )) = ∇ ϕ ( ξ ) 2 L t 2L^{t} 2 L t ω ⊥ \omega^\perp ω ⊥ w 0 > 0 w_0>0 w 0 > 0 B B B ω ⊥ \omega^\perp ω ⊥ ∣ W ∣ ≤ w 0 |W|\leq w_0 ∣ W ∣ ≤ w 0 [ d B ( W ) , ω ] [dB(W),\omega] [ d B ( W ) , ω ]
We use one further complex coordinate Φ \Phi Φ ω \omega ω τ \tau τ P ∈ C k n P\in\mathbb{C}^{n}_{k} P ∈ C k n
P = τ B ( W ) + Φ ω , R 2 = 2 τ 1 + W ⋅ W ‾ , y = ∣ R ∣ 2 ≲ s = ∣ τ ∣ . (101) P=\tau B(W)+\Phi\omega,\qquad R^2=\frac{2\tau}{1+W\cdot\overline{W}},\qquad y=|R|^2\lesssim s=|\tau|. \tag*{(101)} P = τ B ( W ) + Φ ω , R 2 = 1 + W ⋅ W 2 τ , y = ∣ R ∣ 2 ≲ s = ∣ τ ∣. (101) Inverting the projection of P / τ P/\tau P / τ ω ⊥ \omega^\perp ω ⊥ W W W Φ \Phi Φ ω \omega ω W = v ( ξ ) W=v(\xi) W = v ( ξ ) ξ ⋅ B ( W ) \xi\cdot B(W) ξ ⋅ B ( W ) ξ ⋅ B ( W ) = ϕ ( ξ ) \xi\cdot B(W)=\phi(\xi) ξ ⋅ B ( W ) = ϕ ( ξ )
d Φ = ξ ⋅ d P − ϕ ( ξ ) d τ ξ ⋅ ω ( W = v ( ξ ) ) . d\Phi=\frac{\xi\cdot dP-\phi(\xi)\,d\tau}{\xi\cdot\omega}\qquad(W=v(\xi)). d Φ = ξ ⋅ ω ξ ⋅ d P − ϕ ( ξ ) d τ ( W = v ( ξ )) . Let a = K h a=Kh a = K h Q = Φ / a Q=\Phi/a Q = Φ/ a ( T , τ ) (T,\tau) ( T , τ ) § 4 in the real variables, times
r = ∣ W ∣ 2 / w 0 2 + 100 ∣ Q ∣ 2 − 1 < 0. r=|W|^2/w_0^2+100|Q|^2-1<0. r = ∣ W ∣ 2 / w 0 2 + 100∣ Q ∣ 2 − 1 < 0. in the imaginary variables. For a < s < 4 a<s<4 a < s < 4
∣ d W ∣ ≲ s − 1 , ∣ d Φ ∣ ≲ 1 , d P ≍ a 2 s 2 d W d Q . (102) |dW|\lesssim s^{-1},\qquad|d\Phi|\lesssim1,\qquad dP\asymp a^2s^2\,dW\,dQ. \tag*{(102)} ∣ d W ∣ ≲ s − 1 , ∣ d Φ∣ ≲ 1 , d P ≍ a 2 s 2 d W d Q . (102) The derivative bounds include the time variables: solve d P − B d τ = τ d B d W + ω d Φ dP-B\,d\tau=\tau\,dB\,dW+\omega\,d\Phi d P − B d τ = τ d B d W + ω d Φ τ \tau τ ( W , Q ) (W,Q) ( W , Q ) [ τ d B , a ω ] [\tau\,dB,a\omega] [ τ d B , aω ] ∂ ω Φ = 1 \partial_\omega\Phi=1 ∂ ω Φ = 1 F = ( W / w 0 , 10 Q ) F=(W/w_0,10Q) F = ( W / w 0 , 10 Q ) H ( r ) = 4 ( d F ) ∗ d F H(r)=4(dF)^*dF H ( r ) = 4 ( d F ) ∗ d F ≳ s − 2 I \gtrsim s^{-2}I ≳ s − 2 I ≍ a − 2 s − 2 d \asymp a^{-2}s^{-2d} ≍ a − 2 s − 2 d ∣ F ∣ ≤ 1 |F|\leq1 ∣ F ∣ ≤ 1 G f i b ( r ) G f i b ( r ) ∗ ≤ H f i b ( r ) G_{\mathrm{fib}}(r)G_{\mathrm{fib}}(r)^*\leq H_{\mathrm{fib}}(r) G fib ( r ) G fib ( r ) ∗ ≤ H fib ( r ) n n n ker d Φ ∩ G ( r ) ⊥ \ker d\Phi\cap G(r)^\perp ker d Φ ∩ G ( r ) ⊥ d Φ d\Phi d Φ d F dF d F G ( r ) = 2 ( d F ) ∗ F i G(r)=2(dF)^*Fi G ( r ) = 2 ( d F ) ∗ F i ∣ Φ ∣ 2 |\Phi|^2 ∣Φ ∣ 2
The current and its Gaussian multiplier
Take a finite periodic Fourier sum g T = e − i T ϕ ( D ) f g_T=e^{-iT\phi(D)}f g T = e − i T ϕ ( D ) f ∣ ξ ∣ ≍ h − 1 |\xi|\asymp h^{-1} ∣ ξ ∣ ≍ h − 1 ∣ v ( ξ ) ∣ ≤ γ |v(\xi)|\leq\gamma ∣ v ( ξ ) ∣ ≤ γ γ < w 0 / 10 \gamma<w_0/10 γ < w 0 /10 u ξ u_\xi u ξ ∑ ξ ∣ ξ ∣ d / 2 E ( ξ ) u ξ \sum_\xi|\xi|^{d/2}E(\xi)u_\xi ∑ ξ ∣ ξ ∣ d /2 E ( ξ ) u ξ C 0 = I C^0=I C 0 = I C j = C j C^j=C_j C j = C j § 2 , write its paired auxiliary states as w = w 1 + k w 2 = Re ( W ) w=w_1+kw_2=\operatorname{Re}(W) w = w 1 + k w 2 = Re ( W )
Θ ( w , Φ ) = ( 1 2 w t C μ w ) μ = 0 n + ( 0 , ω ) Φ . \Theta(w,\Phi)=\left(\frac{1}{2}w^tC^\mu w\right)_{\mu=0}^{n}+(0,\omega)\Phi. Θ ( w , Φ ) = ( 2 1 w t C μ w ) μ = 0 n + ( 0 , ω ) Φ. Its output is exactly ( τ , P ) (\tau,P) ( τ , P ) M μ a = ( C μ w ) a M_{\mu a}=(C^\mu w)_a M μ a = ( C μ w ) a J = ∣ det [ M , ( 0 , ω ) ] ∣ 2 = ∣ R ∣ 2 n j ( W ) J=|\det[M,(0,\omega)]|^2=|R|^{2n}j(W) J = ∣ det [ M , ( 0 , ω )] ∣ 2 = ∣ R ∣ 2 n j ( W ) j ( W ) ≍ 1 j(W)\asymp1 j ( W ) ≍ 1 n n n R R R
Let G s y s G_{\mathrm{sys}} G sys ( T , X ) − i q R (T,X)-i q_R ( T , X ) − i q R q R μ = w ∗ C μ w / 2 q_R^\mu=w^*C^\mu w/2 q R μ = w ∗ C μ w /2 (9) , the general row formula (10) gives
χ μ = J − 1 / 2 ∑ a M μ a i ( G s y s ) a , A 0 = χ χ ∗ . (103) \chi_\mu=J^{-1/2}\sum_a M_{\mu a}i(G_{\mathrm{sys}})_a,\qquad A^0=\chi\chi^*. \tag*{(103)} χ μ = J − 1/2 a ∑ M μ a i ( G sys ) a , A 0 = χ χ ∗ . (103) The change of variables preserves column divergence: its Hessian term pairs the two equal diagonal state blocks with ( ∂ w 1 a w 1 b + ∂ w 2 a w 2 b ) Θ μ = 0 (\partial_{w_1^a w_1^b}+\partial_{w_2^a w_2^b})\Theta^\mu=0 ( ∂ w 1 a w 1 b + ∂ w 2 a w 2 b ) Θ μ = 0 Φ \Phi Φ d Φ M = 0 \mathrm{d}\Phi\,M=0 d Φ M = 0 d Φ χ = 0 \mathrm{d}\Phi\,\chi=0 d Φ χ = 0 R R R χ \chi χ A 0 A^0 A 0
The attenuation of each plane wave is determined by S ( ξ ) E ( ξ ) = 0 S(\xi)E(\xi)=0 S ( ξ ) E ( ξ ) = 0
exp ( − w ∗ S ( ξ ) w / 2 ) = e − Ψ , Ψ = y 2 h ( W − v ( η ) ) ∗ S ⊥ ( η ) ( W − v ( η ) ) , η = h ξ . (104) \exp(-w^*S(\xi)w/2)=e^{-\Psi},\qquad\Psi=\frac{y}{2h}(W-v(\eta))^*S_\perp(\eta)(W-v(\eta)),\qquad\eta=h\xi. \tag*{(104)} exp ( − w ∗ S ( ξ ) w /2 ) = e − Ψ , Ψ = 2 h y ( W − v ( η ) ) ∗ S ⊥ ( η ) ( W − v ( η )) , η = h ξ . (104) For each row norm at fixed imaginary coordinates we may remove the common left unit R / ∣ R ∣ R/|R| R /∣ R ∣ ∣ R ∣ / J = j ( W ) − 1 / 2 y − d / 2 |R|/\sqrt{J}=j(W)^{-1/2}y^{-d/2} ∣ R ∣/ J = j ( W ) − 1/2 y − d /2 C i \mathbb{C}_i C i k C i k\mathbb{C}_i k C i g T g_T g T i i i C k \mathbb{C}_k C k
j ( W ) − 1 / 2 e ( W ) t C μ E ( η ) ∣ η ∣ d / 2 ( h y ) − d / 2 e − Ψ . (105) j(W)^{-1/2}e(W)^tC^\mu E(\eta)|\eta|^{d/2}(hy)^{-d/2}e^{-\Psi}. \tag*{(105)} j ( W ) − 1/2 e ( W ) t C μ E ( η ) ∣ η ∣ d /2 ( h y ) − d /2 e − Ψ . (105) For the time row the coefficient is j ( W ) − 1 / 2 ( 1 + W ⋅ v ( η ) ) j(W)^{-1/2}(1+W\cdot v(\eta)) j ( W ) − 1/2 ( 1 + W ⋅ v ( η )) A T T 0 ≥ ∣ S 0 g T ∣ 2 A^0_{TT}\geq|S_0g_T|^2 A T T 0 ≥ ∣ S 0 g T ∣ 2 S 0 S_0 S 0 μ = 0 \mu=0 μ = 0
The fixed cap estimate
Normalize ∥ f ∥ 2 = 1 \|f\|_2=1 ∥ f ∥ 2 = 1 σ = s / h \sigma=s/h σ = s / h p = p c p=p_c p = p c q = p / 2 = n / d q=p/2=n/d q = p /2 = n / d J ℓ J_\ell J ℓ g ( Q ) g(Q) g ( Q ) (36) put
A w = g ( Q ) ( J h ∗ X A 0 ) , A = A w + h 50 n C ( H ( r ) ) , μ = A T T , B 0 = λ ∣ Φ ∣ 2 , λ = h − 30 n . A_w=g(Q)(J_h*_{X}A^0),\qquad A=A_w+h^{50n}C(H(r)),\qquad\mu=A_{TT},\qquad B_0=\lambda|\Phi|^2,\qquad\lambda=h^{-30n}. A w = g ( Q ) ( J h ∗ X A 0 ) , A = A w + h 50 n C ( H ( r )) , μ = A T T , B 0 = λ ∣Φ ∣ 2 , λ = h − 30 n . Since Φ \Phi Φ X X X G ( g ( Q ) ) G(g(Q)) G ( g ( Q )) ker H ( r ) ⊂ ker d Φ \ker H(r)\subset\ker\mathrm{d}\Phi ker H ( r ) ⊂ ker d Φ T T TT T T det H f i b ( r ) > 0 \det H_{\mathrm{fib}}(r)>0 det H fib ( r ) > 0 A : H ( B 0 ) = 0 A:H(B_0)=0 A : H ( B 0 ) = 0 μ > 0 \mu>0 μ > 0 H f i b ( B 0 ) H_{\mathrm{fib}}(B_0) H fib ( B 0 ) ≍ λ \asymp\lambda ≍ λ Theorem A.1 gives the solution u ≤ 0 u\leq0 u ≤ 0 (39) , smooth in ( T , τ ) (T,\tau) ( T , τ ) H ( r ) H(r) H ( r ) G ( r ) ⊥ ∩ ker H ( B 0 ) G(r)^\perp\cap\ker H(B_0) G ( r ) ⊥ ∩ ker H ( B 0 ) Proposition 4.2 . That balance and Lemma 4.3 give, for the energy F = J z ( u ) \mathcal{F}=\mathcal{J}_z(u) F = J z ( u ) (42) ,
F ≍ ∫ r < 0 ( − u ) μ d X d P , Δ T , τ F ≤ ∫ ( u / r ) G ( r ) ∗ A w G ( r ) δ ( r ) d X d P . (106) \mathcal{F}\asymp\int_{r<0}(-u)\mu\,\mathrm{d}X\,\mathrm{d}P,\qquad\Delta_{T,\tau}\mathcal{F}\leq\int(u/r)G(r)^*A_wG(r)\,\delta(r)\,\mathrm{d}X\,\mathrm{d}P. \tag*{(106)} F ≍ ∫ r < 0 ( − u ) μ d X d P , Δ T , τ F ≤ ∫ ( u / r ) G ( r ) ∗ A w G ( r ) δ ( r ) d X d P . (106) We check the two Gaussian estimates used in the proof of Proposition 5.1 , throughout 2 a ≤ s ≤ 3 2a\leq s\leq3 2 a ≤ s ≤ 3 ∣ v ∣ < w 0 / 3 |v|<w_0/3 ∣ v ∣ < w 0 /3 ∣ ∂ η Ψ ∣ ≲ α σ |\partial_\eta\Psi|\lesssim_\alpha\sigma ∣ ∂ η Ψ∣ ≲ α σ 2 n + 4 2n+4 2 n + 4 C ( h s ) − d / 2 σ m C(hs)^{-d/2}\sigma^m C ( h s ) − d /2 σ m m m m r = 0 r=0 r = 0 g ( Q ) ≠ 0 g(Q)\ne0 g ( Q ) = 0 ∣ W ∣ ≥ 3 w 0 / 2 |W|\geq\sqrt{3}\,w_0/2 ∣ W ∣ ≥ 3 w 0 /2 S ⊥ ( η ) ≥ c I S_\perp(\eta)\geq cI S ⊥ ( η ) ≥ c I e − c 0 σ e^{-c_0\sigma} e − c 0 σ c 0 > 0 c_0>0 c 0 > 0 γ \gamma γ (50) now gives, for P ∗ = ( J D s ∗ ∣ g T ∣ 2 ) 1 / d P_*=(J_{Ds}*|g_T|^2)^{1/d} P ∗ = ( J D s ∗ ∣ g T ∣ 2 ) 1/ d D D D
μ ≲ s − 2 d σ m P ∗ d ( r ≤ 0 ) , ∥ A w ∥ ≲ s − 2 d σ m e − c 0 σ P ∗ d ( r = 0 ) . (107) \mu\lesssim s^{-2d}\sigma^mP_*^d\quad(r\leq0),\qquad\|A_w\|\lesssim s^{-2d}\sigma^m e^{-c_0\sigma}P_*^d\quad(r=0). \tag*{(107)} μ ≲ s − 2 d σ m P ∗ d ( r ≤ 0 ) , ∥ A w ∥ ≲ s − 2 d σ m e − c 0 σ P ∗ d ( r = 0 ) . (107) The cofactor adds O ( h 50 n σ − 2 s − 2 d ) O(h^{50n}\sigma^{-2}s^{-2d}) O ( h 50 n σ − 2 s − 2 d ) μ \mu μ P ∗ d ≳ h n + 4 P_*^d\gtrsim h^{n+4} P ∗ d ≳ h n + 4 ∥ g T ∥ 2 = 1 \|g_T\|_2=1 ∥ g T ∥ 2 = 1 ∣ W ∣ ≤ γ |W|\leq\gamma ∣ W ∣ ≤ γ ∣ Q ∣ ≤ 1 / 400 |Q|\leq1/400 ∣ Q ∣ ≤ 1/400 h h h ∣ W ∣ ≤ 2 γ |W|\leq2\gamma ∣ W ∣ ≤ 2 γ ∣ Q ∣ ≤ 1 / 100 |Q|\leq1/100 ∣ Q ∣ ≤ 1/100 g = 1 g=1 g = 1 K K K 1 / γ 1/\gamma 1/ γ S 0 S_0 S 0 ∣ v ∣ ≤ 2 γ |v|\leq2\gamma ∣ v ∣ ≤ 2 γ Ψ ≤ C 1 γ 2 σ \Psi\leq C_1\gamma^2\sigma Ψ ≤ C 1 γ 2 σ 2 n + 4 2n+4 2 n + 4 C γ ( h s ) d / 2 σ m e C 1 γ 2 σ C_\gamma(hs)^{d/2}\sigma^m e^{C_1\gamma^2\sigma} C γ ( h s ) d /2 σ m e C 1 γ 2 σ C 1 C_1 C 1 γ \gamma γ J h J_h J h A T 0 ≥ ∣ S 0 g T ∣ 2 A_T^0 \geq|S_0g_T|^2 A T 0 ≥ ∣ S 0 g T ∣ 2 (56) ,
μ ≥ m X = c γ ( h s ) − d σ − m e − 2 C 1 γ 2 σ ∣ g T ( X ) ∣ 2 \mu\geq m_X = c_\gamma(hs)^{-d}\sigma^{-m}e^{-2C_1\gamma^2\sigma}|g_T(X)|^2 μ ≥ m X = c γ ( h s ) − d σ − m e − 2 C 1 γ 2 σ ∣ g T ( X ) ∣ 2 on the whole ball: moving the real coordinate by at most h h h J h J_h J h (52) , H f i b ( r P ∗ ) ≥ P ∗ H f i b ( r ) / 2 H_{\mathrm{fib}}(rP_*) \geq P_*H_{\mathrm{fib}}(r)/2 H fib ( r P ∗ ) ≥ P ∗ H fib ( r ) /2 H f i b ( r ) ≳ s − 2 H_{\mathrm{fib}}(r) \gtrsim s^{-2} H fib ( r ) ≳ s − 2 C σ m ′ r P ∗ C\sigma^{m'}rP_* C σ m ′ r P ∗ 0 ≤ u / r ≲ σ m ′ P ∗ 0 \leq u/r \lesssim\sigma^{m'}P_* 0 ≤ u / r ≲ σ m ′ P ∗ ϵ X 1 / d ( ℓ 2 − h 2 ) \epsilon_X^{1/d}(\ell^2-h^2) ϵ X 1/ d ( ℓ 2 − h 2 ) ℓ \ell ℓ ( − u ) μ ≳ h 2 m X q (-u)\mu\gtrsim h^2m_X^q ( − u ) μ ≳ h 2 m X q m X 1 / d ≤ λ m_X^{1/d} \leq\lambda m X 1/ d ≤ λ O ( h − n ) O(h^{-n}) O ( h − n ) O ( h − n / 2 ) O(h^{-n/2}) O ( h − n /2 ) O ( h − d ) O(h^{-d}) O ( h − d ) μ 1 / d ≤ λ \mu^{1/d} \leq\lambda μ 1/ d ≤ λ h h h
Tangency removes G ( ∣ Q ∣ 2 ) G(|Q|^2) G ( ∣ Q ∣ 2 ) G ( r ) G(r) G ( r ) O ( s − 1 ) O(s^{-1}) O ( s − 1 ) O ( a 2 s 2 d ) O(a^2s^{2d}) O ( a 2 s 2 d ) δ ( r ) \delta(r) δ ( r ) ≍ γ a 2 s 2 d \asymp_\gamma a^2s^{2d} ≍ γ a 2 s 2 d ∫ P ∗ d + 1 d X ≤ ∥ g T ∥ p p \int P_*^{d+1}\,dX \leq\|g_T\|_p^p ∫ P ∗ d + 1 d X ≤ ∥ g T ∥ p p h 2 s 2 d ( h s ) − d q = σ n − 2 h^{2}s^{2d}(hs)^{-dq}=\sigma^{n-2} h 2 s 2 d ( h s ) − d q = σ n − 2
c γ a 2 σ − m e − 2 q C 1 γ 2 σ ∥ g T ∥ p p ≤ F ≤ C a 2 σ m ∥ g T ∥ p p , (108) c_\gamma a^2\sigma^{-m}e^{-2qC_1\gamma^2\sigma}\|g_T\|_p^p \leq\mathcal{F} \leq Ca^2\sigma^m\|g_T\|_p^p, \tag*{(108)} c γ a 2 σ − m e − 2 q C 1 γ 2 σ ∥ g T ∥ p p ≤ F ≤ C a 2 σ m ∥ g T ∥ p p , (108) Δ T , τ F ≤ C a 2 s − 2 σ m e − c 0 σ ∥ g T ∥ p p . (109) \Delta_{T,\tau}\mathcal{F} \leq Ca^2s^{-2}\sigma^m e^{-c_0\sigma}\|g_T\|_p^p. \tag*{(109)} Δ T , τ F ≤ C a 2 s − 2 σ m e − c 0 σ ∥ g T ∥ p p . (109) Choose γ \gamma γ 2 q C 1 γ 2 < c 0 / 2 2qC_1\gamma^2<c_0/2 2 q C 1 γ 2 < c 0 /2 K K K Δ F ≤ C s − 2 e − c s / h F \Delta\mathcal{F}\leq Cs^{-2}e^{-cs/h}\mathcal{F} Δ F ≤ C s − 2 e − cs / h F T = 0 T=0 T = 0 s = 4 a s=4a s = 4 a g 0 = f g_0=f g 0 = f 1 ≤ T ≤ 2 1\leq T\leq2 1 ≤ T ≤ 2 s = 2 a s=2a s = 2 a ∥ g T ∥ p p \|g_T\|_p^p ∥ g T ∥ p p σ \sigma σ b = 2 a b=2a b = 2 a r = 4 a r=4a r = 4 a
∫ 1 2 ∥ g T ∥ p p d T ≲ [ log ( 2 / h ) ] 2 ∥ f ∥ p p . \int_1^2 \|g_T\|_p^p\,dT \lesssim[\log(2/h)]^2\|f\|_p^p. ∫ 1 2 ∥ g T ∥ p p d T ≲ [ log ( 2/ h ) ] 2 ∥ f ∥ p p . Homogeneity removes the normalization.
Finitely many such caps cover the directions in supp b \operatorname{supp} b supp b M ≥ 1 M\geq1 M ≥ 1 sup supp b ∣ ∇ ϕ ∣ \sup_{\operatorname{supp} b}|\nabla\phi| sup supp b ∣∇ ϕ ∣ C / M C/M C / M b M ( η ) = b ( η / M ) b_M(\eta)=b(\eta/M) b M ( η ) = b ( η / M ) ∣ ∇ ( ϕ / M ) ∣ ≤ 1 |\nabla(\phi/M)|\leq1 ∣∇ ( ϕ / M ) ∣ ≤ 1 supp b M \operatorname{supp} b_M supp b M ∣ X ∣ ≥ 3 |X|\geq3 ∣ X ∣ ≥ 3 X − ( T / M ) ∇ ϕ ( η ) X-(T/M)\nabla\phi(\eta) X − ( T / M ) ∇ ϕ ( η ) ≳ ∣ X ∣ \gtrsim|X| ≳ ∣ X ∣ (63) . The cube argument of Section 7 gives the Euclidean estimate for ( C / M , b M ) (C/M,b_M) ( C / M , b M ) f M ( X ) = f ( M X ) f_M(X)=f(MX) f M ( X ) = f ( M X )
e − i T ϕ ( D ) / M b M ( h D ) f M ( X ) = ( e − i T ϕ ( D ) b ( h D ) f ) ( M X ) . e^{-iT\phi(D)/M}b_M(hD)f_M(X)=\left(e^{-iT\phi(D)}b(hD)f\right)(MX). e − i T ϕ ( D ) / M b M ( h D ) f M ( X ) = ( e − i T ϕ ( D ) b ( h D ) f ) ( M X ) . Both norms gain the same factor M − n / p M^{-n/p} M − n / p h h h − C -C − C b ˉ \bar b b ˉ f ( X ) ↦ f ( − X ) f(X)\mapsto f(-X) f ( X ) ↦ f ( − X )
A lower bound on a single patch
For every nonzero b b b 1 / p c 1/p_c 1/ p c b ≠ 0 b\neq0 b = 0 η = ρ ( α , 1 ) \eta=\rho(\alpha,1) η = ρ ( α , 1 ) ρ > 0 \rho>0 ρ > 0 ϕ ( η ) = ρ φ ( α ) \phi(\eta)=\rho\varphi(\alpha) ϕ ( η ) = ρφ ( α ) D 2 φ D^2\varphi D 2 φ ϕ \phi ϕ h = N − 1 h=N^{-1} h = N − 1 f N ^ ( ξ ) = e ( 3 / 2 ) i ϕ ( ξ ) ( ψ / b ) ( ξ / N ) \widehat{f_N}(\xi)=e^{(3/2)i\phi(\xi)}(\psi/b)(\xi/N) f N ( ξ ) = e ( 3/2 ) i ϕ ( ξ ) ( ψ / b ) ( ξ / N ) ψ \psi ψ α \alpha α ∥ f N ∥ ∞ ≲ N n − d / 2 \|f_N\|_\infty\lesssim N^{n-d/2} ∥ f N ∥ ∞ ≲ N n − d /2 X X X ∥ f N ∥ 2 ≲ N n / 2 \|f_N\|_2\lesssim N^{n/2} ∥ f N ∥ 2 ≲ N n /2 ∥ f N ∥ p ≲ N ( n + 1 ) / 2 − 1 / p \|f_N\|_p\lesssim N^{(n+1)/2-1/p} ∥ f N ∥ p ≲ N ( n + 1 ) /2 − 1/ p p ≥ 2 p\geq2 p ≥ 2 3 / 2 + δ 3/2+\delta 3/2 + δ X ′ = δ ∇ φ ( α ) X'=\delta\nabla\varphi(\alpha) X ′ = δ ∇ φ ( α ) X n = δ ( φ ( α ) − α ⋅ ∇ φ ( α ) ) + z / N X_n=\delta(\varphi(\alpha)-\alpha\cdot\nabla\varphi(\alpha))+z/N X n = δ ( φ ( α ) − α ⋅ ∇ φ ( α )) + z / N α \alpha α ∣ z ∣ ≤ c |z|\leq c ∣ z ∣ ≤ c ≍ δ d / N \asymp\delta^d/N ≍ δ d / N α \alpha α N δ N\delta N δ c N n − d / 2 δ − d / 2 cN^{n-d/2}\delta^{-d/2} c N n − d /2 δ − d /2 ∫ e i ρ z ρ d / 2 ψ ( ρ ( α , 1 ) ) d ρ \int e^{i\rho z}\rho^{d/2}\psi(\rho(\alpha,1))\,d\rho ∫ e i ρ z ρ d /2 ψ ( ρ ( α , 1 )) d ρ c c c O ( ( N δ ) − 1 ) O((N\delta)^{-1}) O (( N δ ) − 1 ) [74 Chapter VIII] . The p p p ∫ C / N 1 / 4 δ d − d p / 2 δ d δ \int_{C/N}^{1/4}\delta^{d-dp/2}\delta\,d\delta ∫ C / N 1/4 δ d − d p /2 δ d δ p = p c p=p_c p = p c log N \log N log N
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