Introduction
Interior derivative estimates are a basic tool for proving Liouville theorems for entire solutions of elliptic equations. The harmonic equation provides a simple model. For a positive harmonic function h h h B 2 R ⊂ R d B_{2R} \subset\mathbb{R}^d B 2 R ⊂ R d
sup B R ∣ D log h ∣ ≤ C d R ; (1) \sup_{B_R} \lvert D \log h \rvert\le\frac{C_d}{R}; \tag*{(1)} B R sup ∣ D log h ∣ ≤ R C d ; (1) see Cheng and Yau [19 ] for its Riemannian generalization. For a nonnegative entire harmonic function h h h h + ε h+\varepsilon h + ε ε > 0 \varepsilon> 0 ε > 0 R → ∞ R \to\infty R → ∞ D h = 0 Dh = 0 D h = 0 h h h
For nonlinear equations, the essential issue in proving a Liouville theorem is to obtain interior second-derivative estimates that remain controlled after rescaling.
Real Monge–Ampère and Hessian equations
For the real Monge–Ampère equation det D 2 v = f > 0 \det D^2v = f > 0 det D 2 v = f > 0 [41 ] provides an estimate for the Hessian. In the constant-density case, a standard form for a smooth strictly convex solution with v < 0 v < 0 v < 0 v = 0 v = 0 v = 0
( − v ) ∥ D 2 v ∥ ≤ C , det D 2 v = 1 , (-v)\lVert D^2v\rVert\le C,\qquad\det D^2v = 1, ( − v ) ∥ D 2 v ∥ ≤ C , det D 2 v = 1 , where C C C v v v D v Dv D v [52 Lemma 3.6] . The section geometry and the affine boundary normalization are part of this estimate. Pogorelov’s singular examples in dimensions d ≥ 3 d \ge3 d ≥ 3 [52 Section 3] .
The associated entire-solution theorem is the Jörgens–Calabi–Pogorelov theorem: every smooth convex solution of det D 2 v = 1 \det D^2v = 1 det D 2 v = 1 R d \mathbb{R}^d R d [35 ] proved the two-dimensional case, Calabi [8 ] treated dimensions at most five, and Pogorelov [42 ] proved the general case. The role of normalized sections and interior estimates in this rigidity theorem is reviewed in [52 Theorem 4.4] .
The k k k
σ k ( D 2 v ) = f , σ k ( λ ) = ∑ 1 ≤ i 1 < ⋯ < i k ≤ d λ i 1 ⋯ λ i k , 1 ≤ k ≤ d . \sigma_k(D^2v)=f,\qquad\sigma_k(\lambda)=\sum_{1\le i_1<\cdots<i_k\le d}\lambda_{i_1}\cdots\lambda_{i_k},\qquad1\le k\le d. σ k ( D 2 v ) = f , σ k ( λ ) = 1 ≤ i 1 < ⋯ < i k ≤ d ∑ λ i 1 ⋯ λ i k , 1 ≤ k ≤ d . The elliptic branch is the admissible cone Γ k = { λ : σ j ( λ ) > 0 , 1 ≤ j ≤ k } \Gamma_k=\{\lambda:\sigma_j(\lambda)>0,\ 1\le j\le k\} Γ k = { λ : σ j ( λ ) > 0 , 1 ≤ j ≤ k } k k k k k k Γ k \Gamma_k Γ k [15 ] developed the classical Dirichlet theory for these and more general concave functions of the Hessian eigenvalues. Chou and Wang [17 Theorem 4.1] obtained weighted interior Hessian estimates using an admissible comparison function above the solution. Sheng, Urbas, and Wang [46 ] extended the weighted interior-estimate method to classes of curvature and curvature-quotient equations. For 2 ≤ k ≤ d − 1 2\le k\le d-1 2 ≤ k ≤ d − 1 ( k + 1 ) (k+1) ( k + 1 ) σ k ( D 2 v ) = f ( x , v , D v ) > 0 \sigma_k(D^2v)=f(x,v,Dv)>0 σ k ( D 2 v ) = f ( x , v , D v ) > 0 [38 ] proved the stronger weight-one estimate ( − v ) Δ v ≤ C (-v)\Delta v\le C ( − v ) Δ v ≤ C
These estimates lead to Liouville theorems under positive quadratic lower growth,
v ( x ) ≥ a ∣ x ∣ 2 − b for large ∣ x ∣ , a > 0. v(x)\ge a|x|^2-b\quad\text{for large }|x|,\qquad a>0. v ( x ) ≥ a ∣ x ∣ 2 − b for large ∣ x ∣ , a > 0. Bao, Chen, Guan, and Ji [3 Theorem 3.2] proved quadraticity for smooth convex entire solutions of σ k ( D 2 v ) = 1 \sigma_k(D^2v)=1 σ k ( D 2 v ) = 1 [38 ] obtained the corresponding result under ( k + 1 ) (k+1) ( k + 1 ) 2 ≤ k ≤ d − 1 2\le k\le d-1 2 ≤ k ≤ d − 1 k k k [55 ] assumed two-sided positive quadratic growth. Chu and Dinew [11 ] developed a common framework for several Hessian operators; for σ k \sigma_k σ k σ k + 1 ( D 2 v ) ≥ − A \sigma_{k+1}(D^2v)\ge-A σ k + 1 ( D 2 v ) ≥ − A [60 ] established weighted estimates and rigidity for 3 ≤ k ≤ d − 1 3\le k\le d-1 3 ≤ k ≤ d − 1 D 2 v ≥ − K I d D^2v\ge-KI_d D 2 v ≥ − K I d [51 ] obtained weighted estimates and rigidity from positive quadratic lower growth for the ( d − 1 ) (d-1) ( d − 1 ) d ≥ 3 d\ge3 d ≥ 3 [7 ] extends this lower-growth rigidity to k k k 2 k > d 2k>d 2 k > d 2 ≤ k ≤ d − 1 2\le k\le d-1 2 ≤ k ≤ d − 1
The quadratic Hessian equation σ 2 ( D 2 v ) = f \sigma_2(D^2v)=f σ 2 ( D 2 v ) = f [56 ] proved interior Hessian estimates for f = 1 f=1 f = 1 R 3 \mathbb{R}^3 R 3 [43 ] treated positive smooth variable right-hand sides in dimension three. Guan and Qiu [28 ] proved estimates in all dimensions for convex solutions, and more generally under Γ 2 \Gamma_2 Γ 2 σ 3 ( D 2 v ) ≥ − A \sigma_3(D^2v)\ge-A σ 3 ( D 2 v ) ≥ − A f ≡ 1 f\equiv1 f ≡ 1 [47 ] , and later treated dimension four without convexity [49 ] ; their higher-dimensional estimates in the latter paper require an additional eigenvalue condition.
For the corresponding Liouville problem, Chang and Yuan [20 ] proved quadraticity for entire solutions of σ 2 ( D 2 v ) = 1 \sigma_2(D^2v)=1 σ 2 ( D 2 v ) = 1
D 2 v ≥ ( δ − 2 d ( d − 1 ) ) I d , δ > 0. D^2v\ge\left(\delta-\sqrt{\frac{2}{d(d-1)}}\right)I_d,\qquad\delta>0. D 2 v ≥ ( δ − d ( d − 1 ) 2 ) I d , δ > 0. In particular, their theorem includes convex solutions without growth assumptions. Shankar and Yuan [48 ] later proved the same conclusion under arbitrary fixed semiconvexity D 2 v ≥ − K I d D^2v\ge-KI_d D 2 v ≥ − K I d [18 Theorems 1.3–1.4] proved rigidity for Γ 2 \Gamma_2 Γ 2 R 3 \mathbb{R}^3 R 3 σ 3 ( D 2 v ) ≥ − A \sigma_3(D^2v) \ge-A σ 3 ( D 2 v ) ≥ − A
Several recent preprints advance this theory. Li and Wu [40 ] obtain interior Hessian estimates for σ 2 ( D 2 v ) = 1 \sigma_2(D^2v) = 1 σ 2 ( D 2 v ) = 1 [61 ] obtain interior C 2 , α C^{2,\alpha} C 2 , α [21 ] treat the full Γ 2 \Gamma_2 Γ 2 [62 Theorem 1.1] prove in a recent preprint that every C 2 C^2 C 2 σ k ( D 2 v ) = c 0 > 0 \sigma_k(D^2v) = c_0 > 0 σ k ( D 2 v ) = c 0 > 0 3 ≤ k ≤ d − 1 3 \le k \le d - 1 3 ≤ k ≤ d − 1 [54 ] constructed nonpolynomial entire admissible solutions when k ≥ 2 k \ge2 k ≥ 2 d ≥ 2 k − 1 d \ge2k - 1 d ≥ 2 k − 1
Complex equations and earlier Liouville theorems
For the complex Monge–Ampère equation det ( u i j ˉ ) = f \det(u_{i\bar j}) = f det ( u i j ˉ ) = f f f f density . Bedford and Taylor [6 , 5 ] developed the weak theory for locally bounded plurisubharmonic potentials. Caffarelli, Kohn, Nirenberg, and Spruck [14 ] established smooth Dirichlet solvability on smooth strictly pseudoconvex domains with smooth positive density and smooth boundary values. On bounded real convex domains, Błocki [2 ] proved interior smoothness for the zero-boundary Dirichlet problem under suitable positive-density assumptions. These boundary and geometric hypotheses are essential to the scope of the cited estimates.
For general plurisubharmonic solutions, smooth positive density alone does not imply interior C 2 C^2 C 2 [4 ] . He [30 ] obtained interior Laplacian estimates from L p L^p L p p > n 2 p > n^2 p > n 2 [4 ] proved local regularity for W 2 , p W^{2,p} W 2 , p n ≥ 2 n \ge2 n ≥ 2 p > n ( n − 1 ) p > n(n-1) p > n ( n − 1 ) C 1 , 1 C^{1,1} C 1 , 1 [25 ] obtain C 2 , α C^{2,\alpha} C 2 , α C 1 , 1 C^{1,1} C 1 , 1 [58 ] obtains Hölder estimates for real second derivatives from a bound for the complex Laplacian and corresponding positive Hölder data.
Interior estimates and Liouville theorems also interact in the complex Hessian equations. Hou, Ma, and Wu [34 ] proved a second-order estimate controlled by the square of the gradient on compact Kähler manifolds. Dinew and Kołodziej [23 ] proved local Laplacian estimates under W 2 , p W^{2,p} W 2 , p p > n ( m − 1 ) p > n(m-1) p > n ( m − 1 ) [24 ] states that bounded entire maximal m m m [50 ] established estimates for a broad class of concave elliptic equations on compact Hermitian manifolds using a C C C
For the constant-density complex Monge–Ampère equation, Riebesehl and Schulz [44 ] proved constancy of the complex Hessian under uniform equivalence of the associated Kähler metric with the Euclidean metric. Constancy of the complex Hessian does not by itself make the potential quadratic, since adding the real part of a holomorphic function leaves u i j ˉ u_{i\bar j} u i j ˉ [59 ] proved quadratic rigidity for entire Bedford–Taylor solutions asymptotic to a prescribed positive quadratic solution: for the standard model, u ( z ) = ∣ z ∣ 2 + o ( ∣ z ∣ 2 ) u(z) = |z|^2 + o(|z|^2) u ( z ) = ∣ z ∣ 2 + o ( ∣ z ∣ 2 ) u − ∣ z ∣ 2 u - |z|^2 u − ∣ z ∣ 2
Li and Sheng [39 Theorem 1.2] proved constancy of the complex Hessian under completeness and a positive quadratic lower growth bound, without an upper growth assumption. This also implies quadraticity of the potential: subtracting the constant Hermitian quadratic part leaves a harmonic function bounded below by a negative quadratic polynomial, to which the argument in Section 7.3 applies. Although [39 Corollary 1.3] states quadraticity under two-sided quadratic growth, the upper bound is unnecessary for this consequence. Bing-Long Chen [9 ] obtained a Liouville theorem under completeness and a weaker lower growth condition involving powers of n n n p > 2 n / ( n + 1 ) p > 2n/(n + 1) p > 2 n / ( n + 1 ) [12 ] obtained weighted interior second-derivative estimates under a matrix condition relating ( u i j ) (u_{ij}) ( u ij ) ( u i j ˉ ) (u_{i\bar{j}}) ( u i j ˉ ) [16 Theorem 1.2] proved quadraticity for smooth entire plurisubharmonic solutions of det ( u i j ˉ ) = 1 \det(u_{i\bar{j}}) = 1 det ( u i j ˉ ) = 1 [16 Sections 3–4] .
Related estimates for the homogeneous complex Monge–Ampère equation concern metric lower bounds [31 ] , preservation of convexity [33 ] , and the rank of the complex Hessian [32 ] , under the geometric and boundary assumptions of those problems. For determinant one in complex dimension two, Chen, Hu, and Sheng [13 ] obtain strict convexity of − − u -\sqrt{-u} − − u u = 0 u = 0 u = 0
The main results
We prove an interior estimate for the real Hessian of a plurisubharmonic solution under weak quadratic growth bounds on a sufficiently large ball, for positive Dini-continuous density with Δ R ( f 1 / n ) ≥ − K \Delta_{\mathbb{R}}(f^{1/n}) \geq-K Δ R ( f 1/ n ) ≥ − K Theorem 1.3 . The main application strengthens the quadratic-growth rigidity conjecture of Gábor Székelyhidi: weak quadratic growth forces entire solutions
det ( u i j ˉ ) = 1 on C n (2) \det(u_{i\bar{j}}) = 1 \qquad\text{on } \mathbb{C}^{n} \tag*{(2)} det ( u i j ˉ ) = 1 on C n (2) to be quadratic; see the original AIM workshop problem list [1 Problem 1.4] . Our result applies to locally bounded Bedford–Taylor solutions, without convexity or completeness assumptions. In comparison with Wang’s asymptotic theorem [59 ] , the growth bounds here permit logarithmic changes on both sides and do not prescribe a quadratic model in advance.
For a smooth plurisubharmonic solution, the complex Hessian is nonnegative and its determinant equals f f f f f f
Here are our derivative and matrix conventions. Write z j = x j + i y j z_j = x_j + \mathrm{i}y_j z j = x j + i y j
∂ z j = 1 2 ( ∂ x j − i ∂ y j ) , ∂ z ˉ j = 1 2 ( ∂ x j + i ∂ y j ) . \partial_{z_j} = \frac{1}{2}(\partial_{x_j} - \mathrm{i}\partial_{y_j}), \qquad\partial_{\bar{z}_j} = \frac{1}{2}(\partial_{x_j} + \mathrm{i}\partial_{y_j}). ∂ z j = 2 1 ( ∂ x j − i ∂ y j ) , ∂ z ˉ j = 2 1 ( ∂ x j + i ∂ y j ) . For a twice continuously differentiable real function φ \varphi φ complex Hessian is
D C 2 φ = ( φ i j ˉ ) i , j = 1 n , φ i j ˉ = ∂ z i ∂ z ˉ j φ . D_{\mathbb{C}}^{2}\varphi= (\varphi_{i\bar{j}})_{i,j=1}^{n}, \qquad\varphi_{i\bar{j}} = \partial_{z_i}\partial_{\bar{z}_j}\varphi. D C 2 φ = ( φ i j ˉ ) i , j = 1 n , φ i j ˉ = ∂ z i ∂ z ˉ j φ . We also write φ i = ∂ z i φ \varphi_i=\partial_{z_i}\varphi φ i = ∂ z i φ φ ‾ j = ∂ z ‾ j φ \overline{\varphi}_j=\partial_{\overline{z}_j}\varphi φ j = ∂ z j φ φ i j = ∂ z i ∂ z j φ \varphi_{ij}=\partial_{z_i}\partial_{z_j}\varphi φ ij = ∂ z i ∂ z j φ D R 2 φ D^2_{\mathbb{R}}\varphi D R 2 φ ( x 1 , … , x n , y 1 , … , y n ) (x_1,\ldots,x_n,y_1,\ldots,y_n) ( x 1 , … , x n , y 1 , … , y n ) tr \operatorname{tr} tr φ j j ‾ = 1 4 ( φ x j x j + φ y j y j ) \varphi_{j\overline{j}}=\frac{1}{4}(\varphi_{x_jx_j}+\varphi_{y_jy_j}) φ j j = 4 1 ( φ x j x j + φ y j y j )
Δ R φ = tr D R 2 φ = ∑ j = 1 n ( φ x j x j + φ y j y j ) = 4 tr D C 2 φ . \Delta_{\mathbb{R}}\varphi=\operatorname{tr}D^2_{\mathbb{R}}\varphi=\sum_{j=1}^{n}(\varphi_{x_jx_j}+\varphi_{y_jy_j})=4\operatorname{tr}D^2_{\mathbb{C}}\varphi. Δ R φ = tr D R 2 φ = j = 1 ∑ n ( φ x j x j + φ y j y j ) = 4 tr D C 2 φ . Thus the trace of the complex Hessian, tr D C 2 φ \operatorname{tr}D^2_{\mathbb{C}}\varphi tr D C 2 φ Δ R φ \Delta_{\mathbb{R}}\varphi Δ R φ
For a Hermitian matrix H H H H [ v ] = ∑ i , j H i j ‾ v i v ‾ j H[v]=\sum_{i,j}H_{i\overline{j}}v_i\overline{v}_j H [ v ] = ∑ i , j H i j v i v j H ≥ 0 H\geq0 H ≥ 0 v v v H > 0 H>0 H > 0 v v v H ≤ H ′ H\leq H' H ≤ H ′ H ′ − H ≥ 0 H'-H\geq0 H ′ − H ≥ 0 v T H v v^THv v T H v T T T I d I_d I d d × d d\times d d × d H ∗ H^* H ∗
We specify the weak meaning of the equation and state the main Liouville theorem, the C 2 C^2 C 2
A function is plurisubharmonic if it is upper semicontinuous and subharmonic on every complex line. For u ∈ C 2 u\in C^2 u ∈ C 2 D C 2 u ≥ 0 D^2_{\mathbb{C}}u\geq0 D C 2 u ≥ 0 D C 2 u > 0 D^2_{\mathbb{C}}u>0 D C 2 u > 0
To state the equation without assuming second derivatives, set
ω 0 = − 1 ∂ ∂ ‾ ∣ z ∣ 2 , ρ ( z ) = e + ∣ z ∣ 2 , ψ ( z ) = 1 + log ρ ( z ) . \omega_0=\sqrt{-1}\partial\overline{\partial}|z|^2,\qquad\rho(z)=e+|z|^2,\qquad\psi(z)=1+\log\rho(z). ω 0 = − 1 ∂ ∂ ∣ z ∣ 2 , ρ ( z ) = e + ∣ z ∣ 2 , ψ ( z ) = 1 + log ρ ( z ) . For smooth u u u
− 1 ∂ ∂ ‾ u = − 1 ∑ i , j u i j ‾ d z i ∧ d z ‾ j , ( − 1 ∂ ∂ ‾ u ) n = det D C 2 u ω 0 n . \sqrt{-1}\partial\overline{\partial}u=\sqrt{-1}\sum_{i,j}u_{i\overline{j}}\,dz_i\wedge d\overline{z}_j,\qquad\left(\sqrt{-1}\partial\overline{\partial}u\right)^n=\det D^2_{\mathbb{C}}u\,\omega_0^n. − 1 ∂ ∂ u = − 1 i , j ∑ u i j d z i ∧ d z j , ( − 1 ∂ ∂ u ) n = det D C 2 u ω 0 n . The measure ω 0 n \omega_0^n ω 0 n R 2 n \mathbb{R}^{2n} R 2 n
For locally bounded plurisubharmonic u u u ( − 1 ∂ ∂ ‾ u ) n \left(\sqrt{-1}\partial\overline{\partial}u\right)^n ( − 1 ∂ ∂ u ) n [5 ] . The weak equation ( − 1 ∂ ∂ ‾ u ) n = f ω 0 n \left(\sqrt{-1}\partial\overline{\partial}u\right)^n=f\omega_0^n ( − 1 ∂ ∂ u ) n = f ω 0 n f f f ω 0 n \omega_0^n ω 0 n
We identify C n \mathbb{C}^n C n R 2 n \mathbb{R}^{2n} R 2 n B t = { h ∈ C n : ∣ h ∣ < t } B_t=\{h\in\mathbb{C}^n:|h|<t\} B t = { h ∈ C n : ∣ h ∣ < t }
Theorem 1.1 (A Liouville theorem with a weak quadratic growth condition). Suppose that u u u C n \mathbb{C}^n C n
( − 1 ∂ ∂ ‾ u ) n = ω 0 n on C n (3) \left(\sqrt{-1}\partial\overline{\partial}u\right)^n=\omega_0^n\quad\text{on }\mathbb{C}^n \tag*{(3)} ( − 1 ∂ ∂ u ) n = ω 0 n on C n (3) in the Bedford–Taylor sense. If, for some constants 0 ≤ α ≤ 1 0\leq\alpha\leq1 0 ≤ α ≤ 1 0 < c ≤ 1 0<c\leq1 0 < c ≤ 1 C ≥ 1 C\geq1 C ≥ 1 u u u
c ρ ψ α ≤ u ≤ C ρ ψ 1 − α , ρ = e + ∣ z ∣ 2 , ψ = 1 + log ρ , (4) \frac{c\rho}{\psi^\alpha}\leq u\leq C\rho\psi^{1-\alpha},\qquad\rho=e+|z|^2,\qquad\psi=1+\log\rho, \tag*{(4)} ψ α c ρ ≤ u ≤ C ρ ψ 1 − α , ρ = e + ∣ z ∣ 2 , ψ = 1 + log ρ , (4) then u u u
We call (1.4) the weak quadratic growth condition . In particular, when α = 0 \alpha= 0 α = 0 α = 1 \alpha= 1 α = 1
α = 0 : c ρ ≤ u ≤ C ρ ψ , α = 1 : c ρ / ψ ≤ u ≤ C ρ . \begin{aligned}
\alpha= 0 &: \quad c\rho\le u \le C\rho\psi,\\
\alpha= 1 &: \quad c\rho/\psi\le u \le C\rho.
\end{aligned} α = 0 α = 1 : c ρ ≤ u ≤ C ρ ψ , : c ρ / ψ ≤ u ≤ C ρ . The following C 2 C^2 C 2 f f f
A Dini modulus is a continuous nondecreasing function ω : [ 0 , 1 ] → [ 0 , ∞ ) \omega: [0,1] \to[0,\infty) ω : [ 0 , 1 ] → [ 0 , ∞ ) ω ( 0 ) = 0 \omega(0)=0 ω ( 0 ) = 0
∫ 0 1 ω ( s ) d s s < ∞ . \int_0^1 \omega(s)\frac{ds}{s}<\infty. ∫ 0 1 ω ( s ) s d s < ∞. Its tail is
D ω ( r ) = ω ( r ) + ∫ 0 r ω ( s ) s d s . (5) D_\omega(r)=\omega(r)+\int_0^r\frac{\omega(s)}{s}\,ds. \tag*{(5)} D ω ( r ) = ω ( r ) + ∫ 0 r s ω ( s ) d s . (5) The tail (1.5) determines the initial scale in the perturbation argument of Section 6.
Theorem 1.3 (C 2 C^2 C 2 Fix 0 ≤ α ≤ 1 0 \le\alpha\le1 0 ≤ α ≤ 1 0 < c ≤ 1 0 < c \le1 0 < c ≤ 1 C ≥ 1 C \ge1 C ≥ 1 0 < c 0 ≤ c 1 0 < c_0 \le c_1 0 < c 0 ≤ c 1 K ≥ 0 K \ge0 K ≥ 0 ω \omega ω L 0 ≥ 16 L_0 \ge16 L 0 ≥ 16 H < ∞ H < \infty H < ∞ n n n L ≥ L 0 L \ge L_0 L ≥ L 0 u u u B 2 L B_{2L} B 2 L
( − 1 ∂ ∂ ˉ u ) n = f ω 0 n , c ρ ψ α ≤ u ≤ C ρ ψ 1 − α in B 2 L . (\sqrt{-1}\partial\bar{\partial}u)^n=f\omega_0^n,\qquad\frac{c\rho}{\psi^\alpha}\le u\le C\rho\psi^{1-\alpha}\quad\text{in }B_{2L}. ( − 1 ∂ ∂ ˉ u ) n = f ω 0 n , ψ α c ρ ≤ u ≤ C ρ ψ 1 − α in B 2 L . Suppose that f ∈ C ( B 2 L ) f \in C(B_{2L}) f ∈ C ( B 2 L )
c 0 ≤ f ≤ c 1 , Δ R ( f 1 / n ) ≥ − K in distributions , c_0\le f\le c_1,\qquad\Delta_{\mathbb{R}}(f^{1/n})\ge-K\quad\text{in distributions}, c 0 ≤ f ≤ c 1 , Δ R ( f 1/ n ) ≥ − K in distributions , and
∣ f ( x ) − f ( y ) ∣ ≤ ω ( ∣ x − y ∣ ) whenever ∣ x − y ∣ ≤ 1. |f(x)-f(y)|\le\omega(|x-y|)\quad\text{whenever }|x-y|\le1. ∣ f ( x ) − f ( y ) ∣ ≤ ω ( ∣ x − y ∣ ) whenever ∣ x − y ∣ ≤ 1. Then u ∈ C 2 ( B 1 ) u\in C^2(B_1) u ∈ C 2 ( B 1 ) ∥ u ∥ C 2 ( B 1 ) ≤ H \|u\|_{C^2(B_1)}\le H ∥ u ∥ C 2 ( B 1 ) ≤ H L L L
Corollary 1.4 (Interior smoothness). Under the hypotheses of Theorem 1.3, if f ∈ C ∞ ( B 1 ) f\in C^\infty(B_1) f ∈ C ∞ ( B 1 ) u ∈ C ∞ ( B 1 ) u\in C^\infty(B_1) u ∈ C ∞ ( B 1 ) f ≡ 1 f\equiv1 f ≡ 1
A main ingredient of the proof is the following estimate for the Green-function averaging error. We first prove it for smooth solutions with smooth positive density, then extend it to nonnegative densities and continuous potentials, and finally to locally bounded Bedford–Taylor solutions. We introduce the averaging operator before stating the estimate.
For a locally bounded Borel measurable function φ \varphi φ z z z
S t φ ( z ) = 1 ∣ ∂ B t ∣ ∫ ∂ B t φ ( z + h ) d σ ( h ) , (6) \mathcal{S}_t\varphi(z)=\frac{1}{|\partial B_t|}\int_{\partial B_t}\varphi(z+h)\,d\sigma(h), \tag*{(6)} S t φ ( z ) = ∣ ∂ B t ∣ 1 ∫ ∂ B t φ ( z + h ) d σ ( h ) , (6) B s φ ( z ) = 1 ∣ B s ∣ ∫ B s φ ( z + h ) d h . (7) \mathcal{B}_s\varphi(z)=\frac{1}{|B_s|}\int_{B_s}\varphi(z+h)\,dh. \tag*{(7)} B s φ ( z ) = ∣ B s ∣ 1 ∫ B s φ ( z + h ) d h . (7) Here d σ d\sigma d σ d h dh d h
The operator used in the estimate averages B s φ B_s\varphi B s φ s s s
G t φ ( z ) = 2 t 2 ∫ 0 t s B s φ ( z ) d s . (8) \mathcal{G}_t\varphi(z)=\frac{2}{t^2}\int_0^t sB_s\varphi(z)\,ds. \tag*{(8)} G t φ ( z ) = t 2 2 ∫ 0 t s B s φ ( z ) d s . (8) Its weights are nonnegative and have total mass one, since 2 t − 2 ∫ 0 t s d s = 1 2t^{-2}\int_0^t s\,ds=1 2 t − 2 ∫ 0 t s d s = 1 G t \mathcal{G}_t G t Green-function averaging operator ; its representation by the Green function of a ball is given in § 3.1 . For a solution w w w
Q t = G t w − w . (9) Q_t=\mathcal{G}_tw-w. \tag*{(9)} Q t = G t w − w . (9) A plurisubharmonic function is subharmonic in the 2 n 2n 2 n B s w ( z ) ≥ w ( z ) B_sw(z)\ge w(z) B s w ( z ) ≥ w ( z ) Q t ≥ 0 Q_t\ge0 Q t ≥ 0
Theorem 1.5 (Pseudo-C 2 C^2 C 2 n , c , C , α n,c,C,\alpha n , c , C , α Theorem 1.1 , and F > 0 F>0 F > 0 ε − = 1 / [ 4 ( n + 1 ) ] \varepsilon_- = 1/[4(n+1)] ε − = 1/ [ 4 ( n + 1 )] A 0 ≥ e A_0\ge\sqrt{e} A 0 ≥ e J ≥ 2 J\ge2 J ≥ 2 E < ∞ E<\infty E < ∞ n , c , C , α , F n,c,C,\alpha,F n , c , C , α , F 2 C 2C 2 C ( F 1 / n + 1 ) n (F^{1/n}+1)^n ( F 1/ n + 1 ) n § 4 . These buffers depend only on the stated data. Set
ρ A = A 0 2 + ∣ z ∣ 2 , s = log ρ A , h ( s ) = ( 1 + s ) − α , a = c / 4 , d = a / 4 , \rho_A=A_0^2+|z|^2,\qquad s=\log\rho_A,\qquad h(s)=(1+s)^{-\alpha},\qquad a=c/4,\qquad d=a/4, ρ A = A 0 2 + ∣ z ∣ 2 , s = log ρ A , h ( s ) = ( 1 + s ) − α , a = c /4 , d = a /4 , V = u + J − a ρ A h ( s ) > 0 , s A = log A 0 2 , s L = log ( A 0 2 + L 2 ) , V=u+J-a\rho_Ah(s)>0,\qquad s_A=\log A_0^2,\qquad s_L=\log(A_0^2+L^2), V = u + J − a ρ A h ( s ) > 0 , s A = log A 0 2 , s L = log ( A 0 2 + L 2 ) , q L = [ log 1 + s L 1 + s A ] − 1 , λ L = K ( 1 + s L ) α 8 ( n + 1 ) d . (10) q_L=\left[\log\frac{1+s_L}{1+s_A}\right]^{-1},\qquad\lambda_L=\frac{K(1+s_L)^\alpha}{8(n+1)d}. \tag*{(10)} q L = [ log 1 + s A 1 + s L ] − 1 , λ L = 8 ( n + 1 ) d K ( 1 + s L ) α . (10) For every fixed R ≥ 2 R\ge2 R ≥ 2 L R > R + 1 L_R>R+1 L R > R + 1 R R R L ≥ L R L\ge L_R L ≥ L R u u u B 2 L B_{2L} B 2 L
( − 1 ∂ ∂ ˉ u ) n = f ω 0 n , 0 ≤ f ≤ F , Δ R ( f 1 / n ) ≥ − K (\sqrt{-1}\partial\bar{\partial}u)^n=f\omega_0^n,\qquad0\le f\le F,\qquad\Delta_{\mathbb{R}}(f^{1/n})\ge-K ( − 1 ∂ ∂ ˉ u ) n = f ω 0 n , 0 ≤ f ≤ F , Δ R ( f 1/ n ) ≥ − K in the Bedford–Taylor and distributional senses, respectively, where f f f z ∈ B R z\in B_R z ∈ B R 0 < t ≤ 1 0<t\le1 0 < t ≤ 1
0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) . (11) 0\le Q_t(z)\le2t^2V(z)^{\varepsilon_-}+(\lambda_Lt^2+Eq_L)V(z). \tag*{(11)} 0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) . (11) *If, in addition, f ≥ c 0 > 0 f\ge c_0>0 f ≥ c 0 > 0 ε + = 1 − 1 / ( 2 n ) \varepsilon_+=1-1/(2n) ε + = 1 − 1/ ( 2 n ) A 0 , J , E , L R A_0,J,E,L_R A 0 , J , E , L R n , c , C , α , F , c 0 , K n,c,C,\alpha,F,c_0,K n , c , C , α , F , c 0 , K R R R L R L_R L R
0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε + + E q L V ( z ) . (12) 0\le Q_t(z)\le2t^2V(z)^{\varepsilon_+}+Eq_LV(z). \tag*{(12)} 0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε + + E q L V ( z ) . (12) In both estimates A 0 , J , E A_0,J,E A 0 , J , E
By (1.10), q L q_L q L 1 / log log L 1/\log\log L 1/ log log L λ L \lambda_L λ L K > 0 K>0 K > 0 f ≡ 1 f\equiv1 f ≡ 1 K = 0 K=0 K = 0 λ L = 0 \lambda_L=0 λ L = 0 V V V V V V L L L Q t ≤ A R t 2 + E R q L Q_t\le A_Rt^2+E_Rq_L Q t ≤ A R t 2 + E R q L A R , E R A_R,E_R A R , E R L L L
Corollary 1.6 (Constant-density averaging estimate). Under the hypotheses of Theorem 1.1 , there are constants A 0 , J A_0,J A 0 , J selected as above such that
0 ≤ G t u − u ≤ 2 t 2 V ε − on C n , 0 < t ≤ 1. (13) 0 \leq\mathcal{G}_t u-u \leq2t^2V^{\varepsilon-}\quad\text{on }\mathbb{C}^n,\qquad0<t\leq1. \tag*{(13)} 0 ≤ G t u − u ≤ 2 t 2 V ε − on C n , 0 < t ≤ 1. (13) Consequently u u u is smooth and
tr D C 2 u ≤ 4 ( n + 1 ) V ε − . (14) \operatorname{tr}D_{\mathbb{C}}^2u\leq4(n+1)V^{\varepsilon-}. \tag*{(14)} tr D C 2 u ≤ 4 ( n + 1 ) V ε − . (14) We call Theorem 1.5 a pseudo-C 2 C^2 C 2 t t t L → ∞ L\to\infty L → ∞ (11) with K = 0 K=0 K = 0 z , t z,t z , t t ↓ 0 t\downarrow0 t ↓ 0 (12) instead gives the small remainder needed in Section 6.
Remark 1.7. A lower growth assumption cannot simply be omitted: boundedness below and quadratic upper growth do not imply rigidity. For example, on C 3 \mathbb{C}^3 C 3
w ( z ) = 16 1 / 3 ( ∣ z 1 z 2 ∣ + ∣ z 1 z 3 ∣ + ∣ z 2 z 3 ∣ ) (15) w(z)=16^{1/3}\left(\lvert z_1z_2\rvert+\lvert z_1z_3\rvert+\lvert z_2z_3\rvert\right) \tag*{(15)} w ( z ) = 1 6 1/3 ( ∣ z 1 z 2 ∣ + ∣ z 1 z 3 ∣ + ∣ z 2 z 3 ∣ ) (15) is a continuous nonquadratic plurisubharmonic solution of ( − 1 ∂ ∂ ˉ w ) 3 = ω 0 3 \left(\sqrt{-1}\partial\bar\partial w\right)^3=\omega_0^3 ( − 1 ∂ ∂ ˉ w ) 3 = ω 0 3
0 ≤ w ( z ) ≤ 16 1 / 3 ∣ z ∣ 2 , w ( t , 0 , 0 ) = 0. 0\leq w(z)\leq16^{1/3}\lvert z\rvert^2,\qquad w(t,0,0)=0. 0 ≤ w ( z ) ≤ 1 6 1/3 ∣ z ∣ 2 , w ( t , 0 , 0 ) = 0. For n > 3 n>3 n > 3 ∑ j = 4 n ∣ z j ∣ 2 \sum_{j=4}^n\lvert z_j\rvert^2 ∑ j = 4 n ∣ z j ∣ 2 (15) gives a counterexample on C n \mathbb{C}^n C n (15) ,
w + 16 1 / 3 2 ∣ z ∣ 2 = 16 1 / 3 2 ( ∣ z 1 ∣ + ∣ z 2 ∣ + ∣ z 3 ∣ ) 2 . (16) w+\frac{16^{1/3}}{2}\lvert z\rvert^2=\frac{16^{1/3}}{2}\left(\lvert z_1\rvert+\lvert z_2\rvert+\lvert z_3\rvert\right)^2. \tag*{(16)} w + 2 1 6 1/3 ∣ z ∣ 2 = 2 1 6 1/3 ( ∣ z 1 ∣ + ∣ z 2 ∣ + ∣ z 3 ∣ ) 2 . (16) The right-hand side is convex, since the sum of the moduli is nonnegative and convex. Hence (16) makes w w w
Remark 1.8. Appendix B gives a counterexample with the additional advantage of being smooth everywhere. In particular, it constructs a nonquadratic v ∈ C ∞ ( C 2 ) v\in C^\infty(\mathbb{C}^2) v ∈ C ∞ ( C 2 )
det D C 2 v = 1 , 0 < v ( z ) ≤ 8 ( 1 + ∣ z ∣ 2 ) . \det D_{\mathbb{C}}^2v=1,\qquad0<v(z)\leq8(1+\lvert z\rvert^2). det D C 2 v = 1 , 0 < v ( z ) ≤ 8 ( 1 + ∣ z ∣ 2 ) . Thus the failure of rigidity under quadratic upper growth is not caused by the singularities of (15) . In fact, for every 2 ≤ k ≤ n 2\leq k\leq n 2 ≤ k ≤ n v v v C n \mathbb{C}^n C n
w ( z 1 , … , z n ) = v ( z 1 , z 2 ) + ∑ j = 3 k ∣ z j ∣ 2 , (17) w(z_1,\ldots,z_n)=v(z_1,z_2)+\sum_{j=3}^{k}\lvert z_j\rvert^2, \tag*{(17)} w ( z 1 , … , z n ) = v ( z 1 , z 2 ) + j = 3 ∑ k ∣ z j ∣ 2 , (17) with an empty sum when k = 2 k=2 k = 2 (17) and Proposition B.1, the complex Hessian of w w w k k k 1 1 1 n − k n-k n − k σ k ( D C 2 w ) = 1 \sigma_k(D_{\mathbb{C}}^2w)=1 σ k ( D C 2 w ) = 1 σ j ( D C 2 w ) > 0 \sigma_j(D_{\mathbb{C}}^2w)>0 σ j ( D C 2 w ) > 0 1 ≤ j ≤ k 1\leq j\leq k 1 ≤ j ≤ k w w w Γ k \Gamma_k Γ k (17) and Proposition B.1 also give 0 < w ( z ) ≤ 8 ( 1 + ∣ z ∣ 2 ) 0<w(z)\leq8(1+\lvert z\rvert^2) 0 < w ( z ) ≤ 8 ( 1 + ∣ z ∣ 2 ) [40 Remark 1.2] for σ 2 ( D R 2 v ) = 1 \sigma_2(D_{\mathbb{R}}^2v)=1 σ 2 ( D R 2 v ) = 1
Organization of the paper
Section 2 explains the averaging identities and their relation to Hessian bounds. Section 3 proves the pseudo-C 2 C^2 C 2
From Green-function averages to a Hessian bound
We explain how an estimate for the averaging error supplies the Hessian bound needed for rigidity. The calculation for smooth functions identifies the second-order quantity being controlled; the subsequent discussion explains how to recover the same bound for a weak solution.
For weak quadratic growth, the intermediate estimate (14) is local rather than uniform on the whole space:
0 ≤ tr D C 2 u ≤ 4 ( n + 1 ) V ε − 0 \le\operatorname{tr} D_{\mathbb{C}}^2 u \le4(n+1)V\varepsilon^{-} 0 ≤ tr D C 2 u ≤ 4 ( n + 1 ) V ε − The growth bound (36) controls the right side by a small power of ρ \rho ρ (141) then makes the complex Hessian constant. We therefore begin with G t φ − φ G_t\varphi-\varphi G t φ − φ
The second-order limit of G t G_t G t
We compute the second-order limit of G t φ − φ G_t\varphi-\varphi G t φ − φ S t S_t S t B s B_s B s G t G_t G t (6) , (7) , and (8) . Here tr D C 2 φ = ∑ i φ i i ˉ \operatorname{tr}D_{\mathbb{C}}^2\varphi=\sum_i\varphi_{i\bar{i}} tr D C 2 φ = ∑ i φ i i ˉ
φ ( z + h ) = φ ( z ) + ∇ R φ ( z ) ⋅ h + 1 2 h T D R 2 φ ( z ) h + o ( ∣ h ∣ 2 ) , (18) \varphi(z+h)=\varphi(z)+\nabla_{\mathbb{R}}\varphi(z)\cdot h+\frac{1}{2}h^{\mathsf T}D_{\mathbb{R}}^2\varphi(z)h+o(|h|^2), \tag*{(18)} φ ( z + h ) = φ ( z ) + ∇ R φ ( z ) ⋅ h + 2 1 h T D R 2 φ ( z ) h + o ( ∣ h ∣ 2 ) , (18) where ∇ R φ \nabla_{\mathbb{R}}\varphi ∇ R φ h h h R 2 n \mathbb{R}^{2n} R 2 n B s B_s B s s 2 / ( 2 n + 2 ) s^2/(2n+2) s 2 / ( 2 n + 2 ) (18) , with tr D R 2 φ = Δ R φ \operatorname{tr}D_{\mathbb{R}}^2\varphi=\Delta_{\mathbb{R}}\varphi tr D R 2 φ = Δ R φ
B s φ − φ = s 2 4 ( n + 1 ) Δ R φ + o ( s 2 ) . B_s\varphi-\varphi=\frac{s^2}{4(n+1)}\Delta_{\mathbb{R}}\varphi+o(s^2). B s φ − φ = 4 ( n + 1 ) s 2 Δ R φ + o ( s 2 ) . Integrating this identity with the weights in (8) , and using Δ R φ = 4 tr D C 2 φ \Delta_{\mathbb{R}}\varphi=4\operatorname{tr}D_{\mathbb{C}}^2\varphi Δ R φ = 4 tr D C 2 φ
G t φ − φ = t 2 8 ( n + 1 ) Δ R φ + o ( t 2 ) = t 2 2 ( n + 1 ) tr D C 2 φ + o ( t 2 ) . (19) G_t\varphi-\varphi=\frac{t^2}{8(n+1)}\Delta_{\mathbb{R}}\varphi+o(t^2)=\frac{t^2}{2(n+1)}\operatorname{tr}D_{\mathbb{C}}^2\varphi+o(t^2). \tag*{(19)} G t φ − φ = 8 ( n + 1 ) t 2 Δ R φ + o ( t 2 ) = 2 ( n + 1 ) t 2 tr D C 2 φ + o ( t 2 ) . (19) By (19) , the bound G t φ − φ ≤ Λ t 2 / [ 2 ( n + 1 ) ] G_t\varphi-\varphi\leq\Lambda t^2/[2(n+1)] G t φ − φ ≤ Λ t 2 / [ 2 ( n + 1 )] tr D C 2 φ ≤ Λ \operatorname{tr}D_{\mathbb{C}}^2\varphi\leq\Lambda tr D C 2 φ ≤ Λ 2 ( n + 1 ) / t 2 2(n+1)/t^2 2 ( n + 1 ) / t 2 φ \varphi φ tr D C 2 φ ≤ Λ \operatorname{tr}D_{\mathbb{C}}^2\varphi\leq\Lambda tr D C 2 φ ≤ Λ 0 ≤ D C 2 φ ≤ Λ I n 0\leq D_{\mathbb{C}}^2\varphi\leq\Lambda I_n 0 ≤ D C 2 φ ≤ Λ I n
From the estimates to a Hessian bound
The domain B 2 L B_{2L} B 2 L G t \mathcal{G}_t G t q L → 0 q_L \to0 q L → 0 (10) . For an entire solution with f = 1 f=1 f = 1 λ L \lambda_L λ L (11) , first with L → ∞ L \to\infty L → ∞ z , t z,t z , t (13) .
For a smooth solution, divide (13) by t 2 t^2 t 2 (19) ; this gives (14) . For a Bedford–Taylor solution, first apply Theorem 1.3 and Corollary 1.4 to translates of u u u § 7 . The same expansion then gives (14) pointwise.
At a fixed finite L L L t 2 t^2 t 2 q L / t 2 q_L/t^2 q L / t 2 (12) to make the averaging remainder small before invoking a local perturbation criterion. The order of these two steps is essential because the remainder E q L V E_{q_LV} E q L V (12) is independent of t t t
The interior estimate for a positive density
We first prove Theorem 1.5 for a smooth potential u u u f f f u u u
Since D C 2 u ≥ 0 D^2_{\mathbb{C}}u \ge0 D C 2 u ≥ 0 u u u f 1 / n f^{1/n} f 1/ n Q t = G t u − u Q_t=\mathcal{G}_tu-u Q t = G t u − u Q t Q_t Q t L Q t \mathcal{L}Q_t L Q t L \mathcal{L} L (20) . They will be applied at a maximum of the weighted quotient. Throughout the averaging calculations, z ∈ B L z \in B_L z ∈ B L 0 < t ≤ 1 0<t\le1 0 < t ≤ 1 B L + 1 ⊂ B 2 L B_{L+1}\subset B_{2L} B L + 1 ⊂ B 2 L
The averaging identities
We derive the two inequalities that connect averaging to the determinant equation: a relation between the averaging error and the trace of the averaged Hessian, and a lower bound for the linearized operator applied to that error. These will provide the analytic input for the comparison argument.
We need an operator that uses the determinant equation at the center of each average. With A = D C 2 u A=D^2_{\mathbb{C}}u A = D C 2 u
L φ = tr ( A − 1 D C 2 φ ) , S = tr A − 1 . (20) \mathcal{L}\varphi=\operatorname{tr}\left(A^{-1}D^2_{\mathbb{C}}\varphi\right), \qquad S=\operatorname{tr}A^{-1}. \tag*{(20)} L φ = tr ( A − 1 D C 2 φ ) , S = tr A − 1 . (20) Strict plurisubharmonicity gives A − 1 > 0 A^{-1}>0 A − 1 > 0 L \mathcal{L} L A − 1 A^{-1} A − 1
L u = n , L ∣ z ∣ 2 = S , S ≥ n f − 1 / n . (21) \mathcal{L}u=n,\qquad\mathcal{L}|z|^2=S,\qquad S\ge nf^{-1/n}. \tag*{(21)} L u = n , L ∣ z ∣ 2 = S , S ≥ n f − 1/ n . (21) The last inequality uses det A − 1 = f − 1 \det A^{-1}=f^{-1} det A − 1 = f − 1
To relate tr D C 2 ( G t u ) \operatorname{tr}D^2_{\mathbb{C}}(\mathcal{G}_tu) tr D C 2 ( G t u ) Z t = S t u − u Z_t=\mathcal{S}_tu-u Z t = S t u − u z z z h = s θ h=s\theta h = s θ (6) gives an integral over the unit sphere:
S s u ( z ) = 1 ∣ ∂ B 1 ∣ ∫ ∂ B 1 u ( z + s θ ) d σ ( θ ) . \mathcal{S}_su(z)=\frac{1}{|\partial B_1|}\int_{\partial B_1}u(z+s\theta)\,d\sigma(\theta). S s u ( z ) = ∣ ∂ B 1 ∣ 1 ∫ ∂ B 1 u ( z + s θ ) d σ ( θ ) . Indeed, surface measure contributes the factor s 2 n − 1 s^{2n-1} s 2 n − 1 ∣ ∂ B s ∣ |\partial B_s| ∣ ∂ B s ∣ u u u
d d s S s u ( z ) = 1 ∣ ∂ B 1 ∣ ∫ ∂ B 1 ∇ R u ( z + s θ ) ⋅ θ d σ ( θ ) . \frac{d}{ds}S_su(z)=\frac{1}{|\partial B_1|}\int_{\partial B_1}\nabla_{\mathbb R}u(z+s\theta)\cdot\theta\,d\sigma(\theta). d s d S s u ( z ) = ∣ ∂ B 1 ∣ 1 ∫ ∂ B 1 ∇ R u ( z + s θ ) ⋅ θ d σ ( θ ) . On ∂ B s \partial B_s ∂ B s ν ( h ) = h / s \nu(h)=h/s ν ( h ) = h / s ∂ ν u ( z + h ) = ∇ R u ( z + h ) ⋅ ν ( h ) \partial_\nu u(z+h)=\nabla_{\mathbb R}u(z+h)\cdot\nu(h) ∂ ν u ( z + h ) = ∇ R u ( z + h ) ⋅ ν ( h ) h = s θ h=s\theta h = s θ
d d s S s u ( z ) = 1 ∣ ∂ B s ∣ ∫ ∂ B s ∂ ν u ( z + h ) d σ ( h ) = 1 ∣ ∂ B s ∣ ∫ B s Δ R u ( z + h ) d h . (22) \begin{aligned}
\frac{d}{ds}S_su(z)&=\frac{1}{|\partial B_s|}\int_{\partial B_s}\partial_\nu u(z+h)\,d\sigma(h)\\
&=\frac{1}{|\partial B_s|}\int_{B_s}\Delta_{\mathbb R}u(z+h)\,dh. \tag*{(22)}
\end{aligned} d s d S s u ( z ) = ∣ ∂ B s ∣ 1 ∫ ∂ B s ∂ ν u ( z + h ) d σ ( h ) = ∣ ∂ B s ∣ 1 ∫ B s Δ R u ( z + h ) d h . (22) Because the ball has real dimension 2 n 2n 2 n ∣ ∂ B s ∣ = ( 2 n / s ) ∣ B s ∣ |\partial B_s|=(2n/s)|B_s| ∣ ∂ B s ∣ = ( 2 n / s ) ∣ B s ∣ B s B_s B s [ s / ( 2 n ) ] B s ( Δ R u ) ( z ) [s/(2n)]B_s(\Delta_{\mathbb R}u)(z) [ s / ( 2 n )] B s ( Δ R u ) ( z )
u j j ˉ = 1 4 ( u x j x j + u y j y j ) , Δ R u = 4 ∑ j = 1 n u j j ˉ = 4 tr D C 2 u . u_{j\bar j}=\frac{1}{4}(u_{x_jx_j}+u_{y_jy_j}),\qquad\Delta_{\mathbb R}u=4\sum_{j=1}^{n}u_{j\bar j}=4\operatorname{tr}D_{\mathbb C}^{2}u. u j j ˉ = 4 1 ( u x j x j + u y j y j ) , Δ R u = 4 j = 1 ∑ n u j j ˉ = 4 tr D C 2 u . Thus (3.3) and (1.7) give
d d s S s u = s 2 n B s ( Δ R u ) = 2 s n B s ( tr D C 2 u ) . (23) \frac{d}{ds}S_su=\frac{s}{2n}B_s(\Delta_{\mathbb R}u)=\frac{2s}{n}B_s(\operatorname{tr}D_{\mathbb C}^{2}u). \tag*{(23)} d s d S s u = 2 n s B s ( Δ R u ) = n 2 s B s ( tr D C 2 u ) . (23) As s ↓ 0 s\downarrow0 s ↓ 0 S s u ( z ) → u ( z ) S_su(z)\to u(z) S s u ( z ) → u ( z ) t t t G t = ( 2 / t 2 ) ∫ 0 t s B s d s G_t=(2/t^2)\int_0^t sB_s\,ds G t = ( 2/ t 2 ) ∫ 0 t s B s d s
Z t = t 2 n tr D C 2 ( G t u ) . (24) Z_t=\frac{t^2}{n}\operatorname{tr}D_{\mathbb C}^{2}(G_tu). \tag*{(24)} Z t = n t 2 tr D C 2 ( G t u ) . (24) Here D C 2 ( G t u ) = G t ( D C 2 u ) D_{\mathbb C}^{2}(G_tu)=G_t(D_{\mathbb C}^{2}u) D C 2 ( G t u ) = G t ( D C 2 u ) u u u
Since Δ R u ≥ 0 \Delta_{\mathbb R}u\geq0 Δ R u ≥ 0 S s u S_su S s u s s s u ≤ B s u u\leq B_su u ≤ B s u B s u B_su B s u S r u S_ru S r u 0 < r ≤ s 0<r\leq s 0 < r ≤ s u ≤ B s u ≤ S t u u\leq B_su\leq S_tu u ≤ B s u ≤ S t u 0 < s ≤ t 0<s\leq t 0 < s ≤ t D C 2 u ( z + h ) D_{\mathbb C}^{2}u(z+h) D C 2 u ( z + h )
0 ≤ Q t ≤ Z t , D C 2 ( G t u ) ≥ 0. (25) 0\leq Q_t\leq Z_t,\qquad D_{\mathbb C}^{2}(G_tu)\geq0. \tag*{(25)} 0 ≤ Q t ≤ Z t , D C 2 ( G t u ) ≥ 0. (25) These weights also explain the name of the averaging operator. If Γ t \Gamma_t Γ t − Δ R -\Delta_{\mathbb R} − Δ R B t B_t B t
G t φ ( z ) = 4 n t 2 ∫ B t Γ t ( h ) φ ( z + h ) d h . (26) G_t\varphi(z)=\frac{4n}{t^2}\int_{B_t}\Gamma_t(h)\varphi(z+h)\,dh. \tag*{(26)} G t φ ( z ) = t 2 4 n ∫ B t Γ t ( h ) φ ( z + h ) d h . (26) Its kernel is nonnegative and has total mass one.
To use a maximum principle, we also need a lower bound for L Q t \mathcal{L}Q_t L Q t z + h z+h z + h z z z det A = f \det A=f det A = f L u = n \mathcal{L}u=n L u = n A ( z ) − 1 / 2 A ( z + h ) A ( z ) − 1 / 2 A(z)^{-1/2}A(z+h)A(z)^{-1/2} A ( z ) − 1/2 A ( z + h ) A ( z ) − 1/2
L ( u ( ⋅ + h ) − u ) ( z ) ≥ n f ( z ) 1 / n ( f ( z + h ) 1 / n − f ( z ) 1 / n ) . (27) \mathcal{L}(u(\cdot+h)-u)(z)\geq\frac{n}{f(z)^{1/n}}\left(f(z+h)^{1/n}-f(z)^{1/n}\right). \tag*{(27)} L ( u ( ⋅ + h ) − u ) ( z ) ≥ f ( z ) 1/ n n ( f ( z + h ) 1/ n − f ( z ) 1/ n ) . (27) The coefficients of L \mathcal{L} L z z z h h h
The hypothesis Δ R ( f 1 / n ) ≥ − K \Delta_{\mathbb{R}}(f^{1/n}) \ge-K Δ R ( f 1/ n ) ≥ − K h ↦ f ( z + h ) 1 / n + K ∣ h ∣ 2 / ( 4 n ) h \mapsto f(z+h)^{1/n}+K|h|^2/(4n) h ↦ f ( z + h ) 1/ n + K ∣ h ∣ 2 / ( 4 n ) B s ∣ h ∣ 2 = n s 2 / ( n + 1 ) \mathcal{B}_s|h|^2=ns^2/(n+1) B s ∣ h ∣ 2 = n s 2 / ( n + 1 )
B s ( f 1 / n ) ( z ) − f ( z ) 1 / n ≥ − K s 2 4 ( n + 1 ) . \mathcal{B}_s(f^{1/n})(z)-f(z)^{1/n}\ge-\frac{Ks^2}{4(n+1)}. B s ( f 1/ n ) ( z ) − f ( z ) 1/ n ≥ − 4 ( n + 1 ) K s 2 . Averaging the preceding inequality with the weights 2 s / t 2 2s/t^2 2 s / t 2 s 2 s^2 s 2 t 2 / 2 t^2/2 t 2 /2
L Q t ≥ − n f ( z ) 1 / n K t 2 8 ( n + 1 ) . (28) \mathcal{L}Q_t\ge-\frac{n}{f(z)^{1/n}}\frac{Kt^2}{8(n+1)}. \tag*{(28)} L Q t ≥ − f ( z ) 1/ n n 8 ( n + 1 ) K t 2 . (28) Since K ≥ 0 K\ge0 K ≥ 0 n f − 1 / n ≤ S nf^{-1/n}\le S n f − 1/ n ≤ S
L Q t ≥ − S K t 2 8 ( n + 1 ) . (29) \mathcal{L}Q_t\ge-\frac{SKt^2}{8(n+1)}. \tag*{(29)} L Q t ≥ − 8 ( n + 1 ) S K t 2 . (29) Prepare the comparison function
We now combine (3.5), (3.6), and (3.10) with a comparison function adapted to the lower growth profile. A scalar denominator defines the bounded comparison set and controls the averaging error on its boundary.
Proposition 3.1 (The smooth weak-growth estimates). Fix 0 ≤ α ≤ 1 0\le\alpha\le1 0 ≤ α ≤ 1 c , C , F > 0 c,C,F>0 c , C , F > 0 K ≥ 0 K\ge0 K ≥ 0 u ∈ C ∞ ( B 2 L ) u\in C^\infty(B_{2L}) u ∈ C ∞ ( B 2 L )
det D C 2 u = f , 0 < f ≤ F , Δ R ( f 1 / n ) ≥ − K , c ρ ψ α − 2 ≤ u ≤ C ρ ψ 1 − α + 2 , (30) \det D_{\mathbb{C}}^2u=f,\qquad0<f\le F,\qquad\Delta_{\mathbb{R}}(f^{1/n})\ge-K,\qquad\frac{c\rho}{\psi^\alpha}-2\le u\le C\rho\psi^{1-\alpha}+2, \tag*{(30)} det D C 2 u = f , 0 < f ≤ F , Δ R ( f 1/ n ) ≥ − K , ψ α c ρ − 2 ≤ u ≤ C ρ ψ 1 − α + 2 , (30) where ρ = e + ∣ z ∣ 2 \rho=e+|z|^2 ρ = e + ∣ z ∣ 2 ψ = 1 + log ρ \psi=1+\log\rho ψ = 1 + log ρ A 0 ≥ e A_0\ge\sqrt{e} A 0 ≥ e J > 0 J>0 J > 0 E < ∞ E<\infty E < ∞
ρ A = A 0 2 + ∣ z ∣ 2 , s = log ρ A , h ( s ) = ( 1 + s ) − α , b ( s ) = ( 1 + s ) 1 − α , \rho_A=A_0^2+|z|^2,\qquad s=\log\rho_A,\qquad h(s)=(1+s)^{-\alpha},\qquad b(s)=(1+s)^{1-\alpha}, ρ A = A 0 2 + ∣ z ∣ 2 , s = log ρ A , h ( s ) = ( 1 + s ) − α , b ( s ) = ( 1 + s ) 1 − α , a = c / 4 , d = a / 4 , V = u + J − a ρ A h ( s ) , ε − = 1 4 ( n + 1 ) , ε + = 1 − 1 2 n . (31) a=c/4,\qquad d=a/4,\qquad V=u+J-a\rho_Ah(s),\qquad\varepsilon_-=\frac{1}{4(n+1)},\qquad\varepsilon_+=1-\frac{1}{2n}. \tag*{(31)} a = c /4 , d = a /4 , V = u + J − a ρ A h ( s ) , ε − = 4 ( n + 1 ) 1 , ε + = 1 − 2 n 1 . (31) For s A = log A 0 2 s_A=\log A_0^2 s A = log A 0 2 s L = log ( A 0 2 + L 2 ) s_L=\log(A_0^2+L^2) s L = log ( A 0 2 + L 2 )
q L = [ log 1 + s L 1 + s A ] − 1 , λ L = K 8 ( n + 1 ) d h ( s L ) . (32) q_L=\left[\log\frac{1+s_L}{1+s_A}\right]^{-1},\qquad\lambda_L=\frac{K}{8(n+1)d h(s_L)}. \tag*{(32)} q L = [ log 1 + s A 1 + s L ] − 1 , λ L = 8 ( n + 1 ) d h ( s L ) K . (32) For every fixed r ≥ 2 r\ge2 r ≥ 2 L r L_r L r L ≥ L r L\ge L_r L ≥ L r
0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) , z ∈ B r , 0 < t ≤ 1. (33) 0\le Q_t(z)\le2t^2V(z)^{\varepsilon_-}+(\lambda_Lt^2+Eq_L)V(z),\qquad z\in B_r,\quad0<t\le1. \tag*{(33)} 0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) , z ∈ B r , 0 < t ≤ 1. (33) The constants are independent of L L L inf f \inf f inf f f ≥ c 0 > 0 f\ge c_0>0 f ≥ c 0 > 0
0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε + + E q L V ( z ) , z ∈ B r , 0 < t ≤ 1 , (34) 0\le Q_t(z)\le2t^2V(z)^{\varepsilon_+}+Eq_LV(z),\qquad z\in B_r,\quad0<t\le1, \tag*{(34)} 0 ≤ Q t ( z ) ≤ 2 t 2 V ( z ) ε + + E q L V ( z ) , z ∈ B r , 0 < t ≤ 1 , (34) after increasing L r L_r L r A 0 , J , E , L r A_0,J,E,L_r A 0 , J , E , L r c 0 c_0 c 0 K K K E E E r r r L r L_r L r r r r
We prove the proposition in the following subsections. The fixed offset 2 2 2
A fixed radial normalization
All lower bounds on A 0 A_0 A 0 A 0 A_0 A 0 J = 4 + c A 0 2 / 2 J=4+cA_0^2/2 J = 4 + c A 0 2 /2 x ≥ e x\ge e x ≥ e H ( x ) = x ( 1 + log x ) − α H(x)=x(1+\log x)^{-\alpha} H ( x ) = x ( 1 + log x ) − α 0 ≤ α ≤ 1 0\le\alpha\le1 0 ≤ α ≤ 1
0 < H ′ ( x ) = ( 1 + log x ) − α ( 1 − α 1 + log x ) ≤ 1. (35) 0<H'(x)=(1+\log x)^{-\alpha}\left(1-\frac{\alpha}{1+\log x}\right)\le1. \tag*{(35)} 0 < H ′ ( x ) = ( 1 + log x ) − α ( 1 − 1 + log x α ) ≤ 1. (35) The bound 0 ≤ ρ A − ρ ≤ A 0 2 0\le\rho_A-\rho\le A_0^2 0 ≤ ρ A − ρ ≤ A 0 2 (35) imply H ( ρ A ) ≤ H ( ρ ) + A 0 2 H(\rho_A)\le H(\rho)+A_0^2 H ( ρ A ) ≤ H ( ρ ) + A 0 2 J = 4 + c A 0 2 / 2 J=4+cA_0^2/2 J = 4 + c A 0 2 /2
u + J ≥ c 2 ρ A h ( s ) + 2 , c 4 ρ A h ( s ) + 2 ≤ V ≤ u + J ≤ C 0 ρ A b ( s ) , (36) u+J\ge\frac{c}{2}\rho_Ah(s)+2,\qquad\frac{c}{4}\rho_Ah(s)+2\le V\le u+J\le C_0\rho_Ab(s), \tag*{(36)} u + J ≥ 2 c ρ A h ( s ) + 2 , 4 c ρ A h ( s ) + 2 ≤ V ≤ u + J ≤ C 0 ρ A b ( s ) , (36) where C 0 = C + c / 2 + 6 C_0=C+c/2+6 C 0 = C + c /2 + 6 A 0 A_0 A 0 L L L V ≥ 2 V\ge2 V ≥ 2 s → ∞ s\to\infty s → ∞
log V = s + O ( log ( 1 + s ) ) , V ≥ ( 1 − 2 a c ) ( u + J ) . (37) \log V=s+O(\log(1+s)),\qquad V\ge\left(1-\frac{2a}{c}\right)(u+J). \tag*{(37)} log V = s + O ( log ( 1 + s )) , V ≥ ( 1 − c 2 a ) ( u + J ) . (37) Since ρ A ≥ A 0 2 \rho_A\ge A_0^2 ρ A ≥ A 0 2 (36) also gives inf V → ∞ \inf V\to\infty inf V → ∞ A 0 → ∞ A_0\to\infty A 0 → ∞
For Γ ( g ) = ∑ A i j g i g j \Gamma(g)=\sum A^{ij}g_i g_j Γ ( g ) = ∑ A ij g i g j L ρ A = S \mathcal{L}\rho_A=S L ρ A = S Γ ( ρ A ) ≤ ρ A S \Gamma(\rho_A)\le\rho_A S Γ ( ρ A ) ≤ ρ A S
L ( ρ A h ) = ( h + h ′ ) S + ( h ′ + h ′ ′ ) Γ ( ρ A ) ρ A ≥ ( h + h ′ − ∣ h ′ + h ′ ′ ∣ ) S ≥ 3 4 h S . (38) \begin{aligned}
\mathcal{L}(\rho_Ah)=(h+h')S+(h'+h'')\frac{\Gamma(\rho_A)}{\rho_A} \\
&\ge(h+h'-|h'+h''|)S\ge\frac{3}{4}hS. \tag*{(38)}
\end{aligned} L ( ρ A h ) = ( h + h ′ ) S + ( h ′ + h ′′ ) ρ A Γ ( ρ A ) ≥ ( h + h ′ − ∣ h ′ + h ′′ ∣ ) S ≥ 4 3 h S . (38) once A 0 A_0 A 0 h ′ / h = − α / ( 1 + s ) h'/h=-\alpha/(1+s) h ′ / h = − α / ( 1 + s ) h ′ ′ / h = α ( α + 1 ) / ( 1 + s ) 2 h''/h=\alpha(\alpha+1)/(1+s)^2 h ′′ / h = α ( α + 1 ) / ( 1 + s ) 2 V = u + J − a ρ A h V=u+J-a\rho_Ah V = u + J − a ρ A h L u = n \mathcal{L}u=n L u = n a = 4 d a=4d a = 4 d (38)
L V ≤ n − 3 d h S . (39) \mathcal{L}V\le n-3dhS. \tag*{(39)} L V ≤ n − 3 d h S . (39) All these choices are uniform for 0 ≤ α ≤ 1 0\le\alpha\le1 0 ≤ α ≤ 1
The scalar denominator
We choose a denominator whose linearized Laplacian is negative when S S S (49) below. Fix τ 0 ∈ ( 0 , 1 ) \tau_0\in(0,1) τ 0 ∈ ( 0 , 1 ) w ∈ C ∞ ( [ s A , s L ) ) w\in C^\infty([s_A,s_L)) w ∈ C ∞ ([ s A , s L )) w > 0 w>0 w > 0 w ′ > 0 w'>0 w ′ > 0 k = w ′ / w > 0 k=w'/w>0 k = w ′ / w > 0 σ = σ 0 h ( s ) \sigma=\sigma_0h(s) σ = σ 0 h ( s ) σ 0 > 0 \sigma_0>0 σ 0 > 0 s s s L L L q q q σ 0 ≤ a / 8 \sigma_0\le a/8 σ 0 ≤ a /8 L L L w = w q w=w_q w = w q k k k τ ↦ τ e − τ \tau\mapsto\tau e^{-\tau} τ ↦ τ e − τ ( 0 , τ 0 ) (0,\tau_0) ( 0 , τ 0 )
V = ρ A σ k τ e − τ , 0 < τ < τ 0 . (40) V=\frac{\rho_A\sigma}{k}\tau e^{-\tau},\qquad0<\tau<\tau_0. \tag*{(40)} V = k ρ A σ τ e − τ , 0 < τ < τ 0 . (40) On this set define
G = σ [ ( τ 2 + τ + 1 ) e − τ − 1 ] , D = w V − ρ A w G . (41) G=\sigma[(\tau^2+\tau+1)e^{-\tau}-1],\qquad D=wV-\rho_AwG. \tag*{(41)} G = σ [( τ 2 + τ + 1 ) e − τ − 1 ] , D = w V − ρ A w G . (41) When differentiating D D D V V V ρ A \rho_A ρ A (40) –(41) give
( 1 − k ) w V ≤ D ≤ w V , D V = w ( 1 − k τ ) , (1-k)wV\le D\le wV,\qquad D_V=w(1-k\tau), ( 1 − k ) w V ≤ D ≤ w V , D V = w ( 1 − k τ ) , D V V = − w k 2 ρ A σ ( 1 − τ ) e − τ < 0 , V D V − D = − ρ A w σ χ ( τ ) < 0. (42) D_{VV}=-\frac{wk^2}{\rho_A\sigma(1-\tau)e^{-\tau}}<0,\qquad VD_V-D=-\rho_Aw\sigma\chi(\tau)<0. \tag*{(42)} D V V = − ρ A σ ( 1 − τ ) e − τ w k 2 < 0 , V D V − D = − ρ A w σ χ ( τ ) < 0. (42) where χ ( τ ) = 1 − ( 1 + τ ) e − τ > 0 \chi(\tau)=1-(1+\tau)e^{-\tau}>0 χ ( τ ) = 1 − ( 1 + τ ) e − τ > 0 0 ≤ G ≤ σ τ e − τ 0\le G\le\sigma\tau e^{-\tau} 0 ≤ G ≤ σ τ e − τ 0 ≤ τ ≤ τ 0 < 1 0\le\tau\le\tau_0<1 0 ≤ τ ≤ τ 0 < 1
To express the radial derivatives without singular remainders, set
R D = D ρ A ρ A + D V ρ A 2 − D V V , γ = 1 + k + h ′ h , β = − k − k ′ k , Z = 1 + β τ . (43) R_D=D_{\rho_A\rho_A}+\frac{D^2_{V\rho_A}}{-D_VV},\qquad\gamma=1+k+\frac{h'}{h},\qquad\beta=-k-\frac{k'}{k},\qquad Z=1+\beta\tau. \tag*{(43)} R D = D ρ A ρ A + − D V V D V ρ A 2 , γ = 1 + k + h h ′ , β = − k − k k ′ , Z = 1 + β τ . (43) Primes on γ , β \gamma,\beta γ , β s s s χ \chi χ τ \tau τ
D ρ A w σ = γ χ + Z χ ′ , (44) \frac{D_{\rho_A}}{w\sigma}=\gamma\chi+Z\chi', \tag*{(44)} w σ D ρ A = γ χ + Z χ ′ , (44) D ρ A + ρ A R D w σ = ( γ 2 + γ ′ ) χ + [ ( 2 γ + β ) Z + β ′ τ ] χ ′ + Z 2 χ ′ ′ , (45) \frac{D_{\rho_A}+\rho_A R_D}{w\sigma}=(\gamma^2+\gamma')\chi+\left[(2\gamma+\beta)Z+\beta'\tau\right]\chi'+Z^2\chi'', \tag*{(45)} w σ D ρ A + ρ A R D = ( γ 2 + γ ′ ) χ + [ ( 2 γ + β ) Z + β ′ τ ] χ ′ + Z 2 χ ′′ , (45) ρ A R D w σ = ( γ 2 − γ + γ ′ ) χ + [ ( 2 γ + β − 1 ) Z + β ′ τ ] χ ′ + Z 2 χ ′ ′ . (46) \frac{\rho_A R_D}{w\sigma}=(\gamma^2-\gamma+\gamma')\chi+\left[(2\gamma+\beta-1)Z+\beta'\tau\right]\chi'+Z^2\chi''. \tag*{(46)} w σ ρ A R D = ( γ 2 − γ + γ ′ ) χ + [ ( 2 γ + β − 1 ) Z + β ′ τ ] χ ′ + Z 2 χ ′′ . (46) For verification, fix ζ = D V = w − w ′ τ \zeta=D_V=w-w'\tau ζ = D V = w − w ′ τ τ s = 1 − ( w ′ ′ / w ′ ) τ = Z \tau_s=1-(w''/w')\tau=Z τ s = 1 − ( w ′′ / w ′ ) τ = Z D − ζ V = ρ A w σ χ ( τ ) D-\zeta V=\rho_Aw\sigma\chi(\tau) D − ζ V = ρ A w σ χ ( τ ) (42) . Differentiating this last identity once and twice in ρ A \rho_A ρ A ζ \zeta ζ (44) –(46) .
The identities
χ + 2 χ ′ + χ ′ ′ = 1 , χ ′ + χ ′ ′ = e − τ (47) \chi+2\chi'+\chi''=1,\qquad\chi'+\chi''=e^{-\tau} \tag*{(47)} χ + 2 χ ′ + χ ′′ = 1 , χ ′ + χ ′′ = e − τ (47) show that the right sides of (45) and (46) equal 1 1 1 e − τ e^{-\tau} e − τ ( γ , β , γ ′ , β ′ ) = ( 1 , 0 , 0 , 0 ) (\gamma,\beta,\gamma',\beta')=(1,0,0,0) ( γ , β , γ ′ , β ′ ) = ( 1 , 0 , 0 , 0 ) (45) –(46) , a sufficiently small fixed δ ∈ ( 0 , 1 / 8 ) \delta\in(0,1/8) δ ∈ ( 0 , 1/8 ) (61) below give
D ≥ 1 2 w V , D V > 0 , R D ≥ 0 , D ρ A + ρ A R D D V ≤ 2 σ 0 h . (48) D\ge\frac{1}{2}wV,\qquad D_V>0,\qquad R_D\ge0,\qquad\frac{D_{\rho_A}+\rho_A R_D}{D_V}\le2\sigma_0h. \tag*{(48)} D ≥ 2 1 w V , D V > 0 , R D ≥ 0 , D V D ρ A + ρ A R D ≤ 2 σ 0 h . (48) This conclusion is uniform for 0 ≤ τ ≤ τ 0 0\le\tau\le\tau_0 0 ≤ τ ≤ τ 0 0 < k ≤ δ 0<k\le\delta 0 < k ≤ δ k k k (45) –(46) .
Choose 0 < σ 0 ≤ a / 8 0<\sigma_0\le a/8 0 < σ 0 ≤ a /8 L D \mathcal{L}D L D ∂ V \partial V ∂ V D V V < 0 D_VV<0 D V V < 0 (42) . Since R D ≥ 0 R_D\ge0 R D ≥ 0 (48) and Γ ( ρ A ) ≤ ρ A S \Gamma(\rho_A)\le\rho_AS Γ ( ρ A ) ≤ ρ A S
L D ≤ D V L V + D ρ A S + R D Γ ( ρ A ) ≤ D V L V + ( D ρ A + ρ A R D ) S . \mathcal{L}D\le D_V\mathcal{L}V+D_{\rho_A}S+R_D\Gamma(\rho_A)\le D_V\mathcal{L}V+(D_{\rho_A}+\rho_AR_D)S. L D ≤ D V L V + D ρ A S + R D Γ ( ρ A ) ≤ D V L V + ( D ρ A + ρ A R D ) S . Substitution of (39) and (48) , with d = a / 4 d=a/4 d = a /4
L D ≤ D V ( n − d h S ) , D V D < 1 V . (49) \mathcal{L}D\le D_V(n-dhS),\qquad\frac{D_V}{D}<\frac{1}{V}. \tag*{(49)} L D ≤ D V ( n − d h S ) , D D V < V 1 . (49) The second inequality follows directly from V D V − D < 0 VD_V-D<0 V D V − D < 0 (42) .
Exclude the interior maximum
We first treat the estimate whose constants do not use inf f \inf f inf f B ( V ) = V ϵ − B(V)=V^{\epsilon-} B ( V ) = V ϵ − B L B_L B L
κ 0 = κ 8 ( n + 1 ) , λ L = κ 0 d h ( s L ) , T = n d h , μ = 1 F T n − 1 , H = Q t − λ L t 2 V − 2 t 2 B ( V ) . (50) \kappa_0=\frac{\kappa}{8(n+1)},\qquad\lambda_L=\frac{\kappa_0}{dh(s_L)},\qquad T=\frac{n}{dh},\qquad\mu=\frac{1}{FT^{n-1}},\qquad H=Q_t-\lambda_Lt^2V-2t^2B(V). \tag*{(50)} κ 0 = 8 ( n + 1 ) κ , λ L = d h ( s L ) κ 0 , T = d h n , μ = F T n − 1 1 , H = Q t − λ L t 2 V − 2 t 2 B ( V ) . (50) Where S ≤ T S\le T S ≤ T A − 1 A^{-1} A − 1 μ \mu μ f − 1 ≥ F − 1 f^{-1}\ge F^{-1} f − 1 ≥ F − 1 n − 1 n-1 n − 1 T T T (24) and (25) gives
L Q t = tr ( A − 1 D C 2 G t u ) − n ≥ n μ t 2 Q t − n ( S ≤ T ) . (51) \mathcal{L}Q_t=\operatorname{tr}(A^{-1}D^2_{\mathbb{C}}G_tu)-n\ge\frac{n\mu}{t^2}Q_t-n\qquad(S\le T). \tag*{(51)} L Q t = tr ( A − 1 D C 2 G t u ) − n ≥ t 2 n μ Q t − n ( S ≤ T ) . (51) Increase A 0 A_0 A 0
μ B ( V ) ≥ 1 , μ V ≥ 2 , B ′ ( V ) ≤ 1 2 . (52) \mu B(V) \ge1,\qquad\mu V \ge2,\qquad B'(V) \le\frac{1}{2}. \tag*{(52)} μ B ( V ) ≥ 1 , μ V ≥ 2 , B ′ ( V ) ≤ 2 1 . (52) This is possible independently of L L L (36) , h n − 1 V ε − h^{n-1}V^{\varepsilon_-} h n − 1 V ε − h n − 1 V h^{n-1}V h n − 1 V s A → ∞ s_A \to\infty s A → ∞ B ′ ( V ) = ε − V ε − − 1 B'(V)=\varepsilon_-V^{\varepsilon_- -1} B ′ ( V ) = ε − V ε − − 1
On S > T S>T S > T T T T (50) and (39) give L V < 0 \mathcal{L}V<0 L V < 0 B B B L B ≤ B ′ L V ≤ 0 \mathcal{L}B\le B'\mathcal{L}V\le0 L B ≤ B ′ L V ≤ 0 (29) and (50) , we obtain
L H ≥ t 2 [ ( 3 d h λ L − κ 0 ) S − n λ L ] ≥ 0. (53) \mathcal{L}H \ge t^2\left[(3dh\lambda_L-\kappa_0)S-n\lambda_L\right]\ge0. \tag*{(53)} L H ≥ t 2 [ ( 3 d h λ L − κ 0 ) S − n λ L ] ≥ 0. (53) Indeed h ≥ h ( s L ) h\ge h(s_L) h ≥ h ( s L ) B L B_L B L κ 0 ≤ d h λ L \kappa_0\le dh\lambda_L κ 0 ≤ d h λ L (53) is at least λ L ( 2 d h S − n ) ≥ 0 \lambda_L(2dhS-n)\ge0 λ L ( 2 d h S − n ) ≥ 0 (49) gives L D < 0 \mathcal{L}D<0 L D < 0
On S ≤ T S\le T S ≤ T (39) and concavity of B B B L V ≤ n \mathcal{L}V\le n L V ≤ n L B ≤ n B ′ \mathcal{L}B\le nB' L B ≤ n B ′ H H H (50) , followed by (51) , yields
L H ≥ n μ t 2 H + n ( 2 μ B − 1 − 2 t 2 B ′ ) + n λ L ( μ V − t 2 ) ≥ n μ t 2 H . (54) \mathcal{L}H \ge\frac{n\mu}{t^2}H+n(2\mu B-1-2t^2B')+n\lambda_L(\mu V-t^2)\ge\frac{n\mu}{t^2}H. \tag*{(54)} L H ≥ t 2 n μ H + n ( 2 μ B − 1 − 2 t 2 B ′ ) + n λ L ( μ V − t 2 ) ≥ t 2 n μ H . (54) The last inequality uses (52) and 0 < t ≤ 1 0<t\le1 0 < t ≤ 1 H / D H/D H / D S ≤ T S\le T S ≤ T (49) would give
L H H ≤ L D D ≤ n D V D < n V < n μ t 2 . (55) \frac{\mathcal{L}H}{H}\le\frac{\mathcal{L}D}{D}\le\frac{nD_V}{D}<\frac{n}{V}<\frac{n\mu}{t^2}. \tag*{(55)} H L H ≤ D L D ≤ D n D V < V n < t 2 n μ . (55) Here the last inequality uses μ V ≥ 2 \mu V\ge2 μ V ≥ 2 (52) and t ≤ 1 t\le1 t ≤ 1 (54) . On S > T S>T S > T (53) and (49) give the contradiction directly. Thus H / D H/D H / D
The refinement for a positive lower density bound
Suppose f ≥ c 0 > 0 f\ge c_0>0 f ≥ c 0 > 0 (28) gives the stronger source bound
L Q t ≥ − κ t 2 , κ = n K 8 ( n + 1 ) c 0 1 / n . (56) \mathcal{L}Q_t\ge-\kappa t^2,\qquad\kappa=\frac{nK}{8(n+1)c_0^{1/n}}. \tag*{(56)} L Q t ≥ − κ t 2 , κ = 8 ( n + 1 ) c 0 1/ n n K . (56) In this paragraph put
B ( V ) = V ε + , T = 2 ( n + κ / B ′ ( V ) ) d h , μ = 1 F T n n − 1 , H = Q t − 2 t 2 B ( V ) . (57) B(V)=V^{\varepsilon_+},\qquad T=\frac{2(n+\kappa/B'(V))}{dh},\qquad\mu=\frac{1}{F T n^{n-1}},\qquad H=Q_t-2t^2B(V). \tag*{(57)} B ( V ) = V ε + , T = d h 2 ( n + κ / B ′ ( V )) , μ = F T n n − 1 1 , H = Q t − 2 t 2 B ( V ) . (57) On S > T S>T S > T (39) and concavity give L B ≤ B ′ ( n − d h S ) \mathcal{L}B\le B'(n-dhS) L B ≤ B ′ ( n − d h S ) T T T (57) , together with (56) and (49) , therefore yields
L H ≥ t 2 [ − κ + 2 B ′ ( d h S − n ) ] ≥ 0 , L D < 0. \mathcal{L}H\ge t^2[-\kappa+2B'(dhS-n)]\ge0,\qquad\mathcal{L}D<0. L H ≥ t 2 [ − κ + 2 B ′ ( d h S − n )] ≥ 0 , L D < 0. On S ≤ T S\le T S ≤ T (51) applies to the new values of T , μ T,\mu T , μ L B ≤ n B ′ \mathcal{L}B\le nB' L B ≤ n B ′ H H H (57) , we obtain
L H ≥ n μ t 2 H + n ( 2 μ B − 1 − 2 t 2 B ′ ) ≥ n μ t 2 H , (58) \mathcal{L}H\ge\frac{n\mu}{t^2}H+n(2\mu B-1-2t^2B')\ge\frac{n\mu}{t^2}H, \tag*{(58)} L H ≥ t 2 n μ H + n ( 2 μ B − 1 − 2 t 2 B ′ ) ≥ t 2 n μ H , (58) provided (52) holds for this corrector.
To verify (52) for the positive-density corrector, we increase A 0 A_0 A 0 (57) gives
μ B = 1 F ( d h 2 ) n − 1 V ε + [ n + ( κ / ε + ) V 1 − ε + ] n − 1 . \mu B=\frac{1}{F}\left(\frac{dh}{2}\right)^{n-1}\frac{V^{\varepsilon_+}}{\left[n+\left(\kappa/\varepsilon_+\right)V^{1-\varepsilon_+}\right]^{n-1}}. μ B = F 1 ( 2 d h ) n − 1 [ n + ( κ / ε + ) V 1 − ε + ] n − 1 V ε + . The quotient is increasing in V V V V V V n ε + − n + 1 = 1 / 2 n\varepsilon_+-n+1=1/2 n ε + − n + 1 = 1/2 (36) therefore makes μ B → ∞ \mu B\to\infty μ B → ∞ s A → ∞ s_A\to\infty s A → ∞ μ V → ∞ \mu V\to\infty μ V → ∞ B ′ = ε + V ε + − 1 → 0 B'=\varepsilon_+V^{\varepsilon_+-1}\to0 B ′ = ε + V ε + − 1 → 0 (52) holds, and the argument in (55) , with (58) , excludes a positive interior maximum for this quotient as well. We now fix one A 0 , J A_0,J A 0 , J c 0 c_0 c 0
Estimate the boundary contribution
We construct weights for which the comparison set is bounded before the spatial boundary is reached. Choose σ 0 \sigma_0 σ 0
a 0 = σ 0 τ 0 e − τ 0 , a 0 δ < c 8 . (59) a_0=\sigma_0\tau_0e^{-\tau_0},\qquad\frac{a_0}{\delta}<\frac{c}{8}. \tag*{(59)} a 0 = σ 0 τ 0 e − τ 0 , δ a 0 < 8 c . (59) For q = q L q=q_L q = q L (32) , define on s A ≤ s < s L s_A\le s<s_L s A ≤ s < s L
I ( s ) = log 1 + s 1 + s A , w q ( s ) = 1 1 − q I ( s ) , k q ( s ) = q w q ( s ) 1 + s . (60) I(s)=\log\frac{1+s}{1+s_A},\qquad w_q(s)=\frac{1}{1-qI(s)},\qquad k_q(s)=\frac{qw_q(s)}{1+s}. \tag*{(60)} I ( s ) = log 1 + s A 1 + s , w q ( s ) = 1 − q I ( s ) 1 , k q ( s ) = 1 + s q w q ( s ) . (60) For s A ≤ s < s L s_A\le s<s_L s A ≤ s < s L (32) and (60) give 1 − q I ( s ) > 0 1-qI(s)>0 1 − q I ( s ) > 0 w q w_q w q [ s A , s L ) [s_A,s_L) [ s A , s L )
w q ′ ( s ) = q w q ( s ) 2 1 + s > 0 , w q ′ w q = k q . w'_q(s)=\frac{qw_q(s)^2}{1+s}>0,\qquad\frac{w'_q}{w_q}=k_q. w q ′ ( s ) = 1 + s q w q ( s ) 2 > 0 , w q w q ′ = k q . The weight w q w_q w q s = s L s=s_L s = s L q L → 0 q_L\to0 q L → 0 L L L k q ( s A ) = q / ( 1 + s A ) < δ k_q(s_A)=q/(1+s_A)<\delta k q ( s A ) = q / ( 1 + s A ) < δ k q ( s ) → ∞ k_q(s)\to\infty k q ( s ) → ∞ s ↑ s L s\uparrow s_L s ↑ s L s 1 < s L s_1<s_L s 1 < s L k q = δ k_q=\delta k q = δ
The coefficient conditions required for (48) are uniform in q q q [ s A , s 1 ] [s_A,s_1] [ s A , s 1 ] r ( s ) = ( 1 + s ) − 1 r(s)=(1+s)^{-1} r ( s ) = ( 1 + s ) − 1 v ( s ) = − α / ( 1 + s ) v(s)=-\alpha/(1+s) v ( s ) = − α / ( 1 + s )
k q ′ = k q ( k q − r ) , γ = 1 + k q + v , β = r − 2 k q , k'_q=k_q(k_q-r),\qquad\gamma=1+k_q+v,\qquad\beta=r-2k_q, k q ′ = k q ( k q − r ) , γ = 1 + k q + v , β = r − 2 k q , γ ′ = k q ′ + v ′ , β ′ = r ′ − 2 k q ′ . (61) \gamma'=k'_q+v',\qquad\beta'=r'-2k'_q. \tag*{(61)} γ ′ = k q ′ + v ′ , β ′ = r ′ − 2 k q ′ . (61) Thus ∣ γ − 1 ∣ + ∣ β ∣ ≤ 3 δ + O ( ( 1 + s A ) − 1 ) |\gamma-1|+|\beta|\le3\delta+O((1+s_A)^{-1}) ∣ γ − 1∣ + ∣ β ∣ ≤ 3 δ + O (( 1 + s A ) − 1 ) ∣ γ ′ ∣ + ∣ β ′ ∣ = O ( δ 2 + ( 1 + s A ) − 2 ) |\gamma'|+|\beta'|=O(\delta^2+(1+s_A)^{-2}) ∣ γ ′ ∣ + ∣ β ′ ∣ = O ( δ 2 + ( 1 + s A ) − 2 ) δ \delta δ A 0 A_0 A 0 (45) –(47) give (48) throughout the stopping interval.
Define the bounded open set
Ω q = { z : s ( z ) < s 1 , V ( z ) < a 0 ρ A h ( s ) k q ( s ) } ⊂ B L . (62) \Omega_q=\left\{z:s(z)<s_1,\ V(z)<\frac{a_0\rho_Ah(s)}{k_q(s)}\right\}\subset B_L. \tag*{(62)} Ω q = { z : s ( z ) < s 1 , V ( z ) < k q ( s ) a 0 ρ A h ( s ) } ⊂ B L . (62) The increasing branch (40) is defined there with 0 < τ < τ 0 0<\tau<\tau_0 0 < τ < τ 0 (36) and (59) , this set is empty near the outer sphere s = s 1 s=s_1 s = s 1
V = a 0 ρ A h k q , τ = τ 0 . (63) V=\frac{a_0\rho_Ah}{k_q},\qquad\tau=\tau_0. \tag*{(63)} V = k q a 0 ρ A h , τ = τ 0 . (63) The rough average bound needed on this boundary is
0 ≤ Q t [ u ] = Q t [ u + J ] ≤ G t ( u + J ) ≤ C 1 ρ A b ( s ) ( z ∈ B L , 0 < t ≤ 1 ) . (64) 0\le Q_t[u]=Q_t[u+J]\le G_t(u+J)\le C_1\rho_A b(s)\qquad(z\in B_L,\ 0<t\le1). \tag*{(64)} 0 ≤ Q t [ u ] = Q t [ u + J ] ≤ G t ( u + J ) ≤ C 1 ρ A b ( s ) ( z ∈ B L , 0 < t ≤ 1 ) . (64) Indeed u + J ≥ 2 u+J\ge2 u + J ≥ 2 B L + 1 B_{L+1} B L + 1 ρ A b ( log ρ A ) \rho_A b(\log\rho_A) ρ A b ( log ρ A )
For either corrector, H ≤ Q t H\le Q_t H ≤ Q t
H D ≤ Q t D ≤ 2 C 1 a 0 ( 1 + s ) k q w q = 2 C 1 a 0 q . (65) \frac{H}{D}\le\frac{Q_t}{D}\le\frac{2C_1}{a_0}\frac{(1+s)k_q}{w_q}=\frac{2C_1}{a_0}q. \tag*{(65)} D H ≤ D Q t ≤ a 0 2 C 1 w q ( 1 + s ) k q = a 0 2 C 1 q . (65) The denominator extends continuously to this boundary since τ 0 < 1 \tau_0<1 τ 0 < 1
Q t ≤ 2 t 2 V ε − + λ L t 2 V + C ∗ q D q , in the general case , Q t ≤ 2 t 2 V ε + + C ∗ q D q , if f ≥ c 0 > 0 , (66) \begin{aligned}
Q_t&\le2t^2V^{\varepsilon_-}+\lambda_Lt^2V+C_*qD_q,\quad&&\text{in the general case},\\
Q_t&\le2t^2V^{\varepsilon_+}+C_*qD_q, &&\text{if }f\ge c_0>0, \tag*{(66)}
\end{aligned} Q t Q t ≤ 2 t 2 V ε − + λ L t 2 V + C ∗ q D q , ≤ 2 t 2 V ε + + C ∗ q D q , in the general case , if f ≥ c 0 > 0 , (66) on Ω q \Omega_q Ω q C ∗ = 2 C 1 / a 0 C_*=2C_1/a_0 C ∗ = 2 C 1 / a 0 q q q L L L r r r ∂ Ω q \partial\Omega_q ∂ Ω q H / D H/D H / D
Fix r ≥ 2 r\ge2 r ≥ 2 q L → 0 q_L\to0 q L → 0 w q L → 1 w_{q_L}\to1 w q L → 1 k q L → 0 k_{q_L}\to0 k q L → 0 B ‾ r \overline{B}_r B r L → ∞ L\to\infty L → ∞ L r L_r L r
q L I ( log ( A 0 2 + r 2 ) ) ≤ 1 2 , 2 C 0 q L < a 0 , 2 q L 1 + s A < δ . q_L I\left(\log(A_0^2+r^2)\right)\le\frac{1}{2},\qquad2C_0q_L<a_0,\qquad\frac{2q_L}{1+s_A}<\delta. q L I ( log ( A 0 2 + r 2 ) ) ≤ 2 1 , 2 C 0 q L < a 0 , 1 + s A 2 q L < δ . These inequalities, together with (3.17) and (3.41), imply B r ⊂ Ω q B_r\subset\Omega_q B r ⊂ Ω q D q D_q D q D q ≤ w q V ≤ 2 V D_q\le w_qV\le2V D q ≤ w q V ≤ 2 V B r B_r B r E = 2 C ∗ E=2C_* E = 2 C ∗
Remark 3.2 (The domain needed by the smooth argument). The proof uses the equation only on B L + 1 B_{L+1} B L + 1 B 3 L / 2 B_{3L/2} B 3 L /2 B L + 1 B_{L+1} B L + 1
Remark 3.3 (The role of the density error). The coefficient λ L \lambda_L λ L L L L K > 0 K>0 K > 0 C 2 C^2 C 2 t 2 t^2 t 2 L L L K = 0 K=0 K = 0 λ L = 0 \lambda_L=0 λ L = 0 ε − \varepsilon_- ε −
Nonnegative densities and continuous potentials
We extend the pseudo-estimate to nonnegative densities and continuous potentials. A comparison estimate first controls the change in a Dirichlet solution in terms of its boundary values and density root. It then gives uniform approximation by smooth solutions with positive densities, while preserving the growth bounds needed for Theorem 1.5.
Comparison of Dirichlet solutions
The matrix inequality behind the comparison is
det ( A + d I ) 1 / n ≥ det ( A ) 1 / n + d ( A ≥ 0 , d ≥ 0 ) . (67) \det(A+dI)^{1/n}\ge\det(A)^{1/n}+d\qquad(A\ge0,\ d\ge0). \tag*{(67)} det ( A + d I ) 1/ n ≥ det ( A ) 1/ n + d ( A ≥ 0 , d ≥ 0 ) . (67) It follows from concavity and homogeneity of det 1 / n \det^{1/n} det 1/ n [22 ] provides the corresponding statement for bounded plurisubharmonic functions, which we use next.
Lemma 4.1 (Stability under changes in the density g i n g_i^n g i n u 1 , u 2 u_1,u_2 u 1 , u 2 B R ‾ \overline{B_R} B R B R B_R B R
( − 1 ∂ ∂ ‾ u i ) n = g i n ω 0 n in B R , g i ∈ C ( B R ‾ ) , g i ≥ 0. (\sqrt{-1}\partial\overline{\partial}u_i)^n=g_i^n\omega_0^n\quad\text{in }B_R,\qquad g_i\in C(\overline{B_R}),\qquad g_i\geq0. ( − 1 ∂ ∂ u i ) n = g i n ω 0 n in B R , g i ∈ C ( B R ) , g i ≥ 0. Set
b = ∥ u 1 − u 2 ∥ L ∞ ( ∂ B R ) , d = ∥ g 1 − g 2 ∥ L ∞ ( B R ) . (68) b=\lVert u_1-u_2\rVert_{L^\infty(\partial B_R)},\qquad d=\lVert g_1-g_2\rVert_{L^\infty(B_R)}. \tag*{(68)} b = ∥ u 1 − u 2 ∥ L ∞ ( ∂ B R ) , d = ∥ g 1 − g 2 ∥ L ∞ ( B R ) . (68) Then
∣ u 1 ( z ) − u 2 ( z ) ∣ ≤ b + d ( R 2 − ∣ z ∣ 2 ) for z ∈ B R ‾ . (69) \lvert u_1(z)-u_2(z)\rvert\leq b+d(R^2-\lvert z\rvert^2)\quad\text{for }z\in\overline{B_R}. \tag*{(69)} ∣ u 1 ( z ) − u 2 ( z )∣ ≤ b + d ( R 2 − ∣ z ∣ 2 ) for z ∈ B R . (69) In particular, ∥ u 1 − u 2 ∥ L ∞ ( B R ) ≤ b + R 2 d \lVert u_1-u_2\rVert_{L^\infty(B_R)}\leq b+R^2d ∥ u 1 − u 2 ∥ L ∞ ( B R ) ≤ b + R 2 d
Proof. To compare u 1 u_1 u 1 u 2 u_2 u 2
v 1 = u 1 − b + d ( ∣ z ∣ 2 − R 2 ) . (70) v_1=u_1-b+d(\lvert z\rvert^2-R^2). \tag*{(70)} v 1 = u 1 − b + d (∣ z ∣ 2 − R 2 ) . (70) Since u 1 u_1 u 1 d ≥ 0 d\geq0 d ≥ 0 v 1 v_1 v 1 [22 Theorem 1.2] , applied to u 1 u_1 u 1 d ∣ z ∣ 2 d\lvert z\rvert^2 d ∣ z ∣ 2
( − 1 ∂ ∂ ‾ v 1 ) n ≥ ( g 1 + d ) n ω 0 n ≥ g 2 n ω 0 n = ( − 1 ∂ ∂ ‾ u 2 ) n . (71) (\sqrt{-1}\partial\overline{\partial}v_1)^n\geq(g_1+d)^n\omega_0^n\geq g_2^n\omega_0^n=(\sqrt{-1}\partial\overline{\partial}u_2)^n. \tag*{(71)} ( − 1 ∂ ∂ v 1 ) n ≥ ( g 1 + d ) n ω 0 n ≥ g 2 n ω 0 n = ( − 1 ∂ ∂ u 2 ) n . (71) By (4.4) and (4.2), v 1 = u 1 − b ≤ u 2 v_1=u_1-b\leq u_2 v 1 = u 1 − b ≤ u 2 ∂ B R \partial B_R ∂ B R [5 ] , with (4.5), therefore yields v 1 ≤ u 2 v_1\leq u_2 v 1 ≤ u 2 B R B_R B R u 1 − u 2 ≤ b + d ( R 2 − ∣ z ∣ 2 ) u_1-u_2\leq b+d(R^2-\lvert z\rvert^2) u 1 − u 2 ≤ b + d ( R 2 − ∣ z ∣ 2 ) u 1 u_1 u 1 u 2 u_2 u 2
We will also need continuous Dirichlet solutions when the prescribed root vanishes. Lemma 4.1 constructs them as uniform limits of smooth positive-density solutions.
Lemma 4.2 (Continuous Dirichlet solutions). Let g ∈ C ( B R ‾ ) g\in C(\overline{B_R}) g ∈ C ( B R ) η ∈ C ∞ ( ∂ B R ) \eta\in C^\infty(\partial B_R) η ∈ C ∞ ( ∂ B R ) v ∈ C ( B R ‾ ) v\in C(\overline{B_R}) v ∈ C ( B R ) B R B_R B R
( − 1 ∂ ∂ ‾ v ) n = g n ω 0 n in B R , v = η on ∂ B R . (\sqrt{-1}\partial\overline{\partial}v)^n=g^n\omega_0^n\quad\text{in }B_R,\qquad v=\eta\quad\text{on }\partial B_R. ( − 1 ∂ ∂ v ) n = g n ω 0 n in B R , v = η on ∂ B R . Proof. Choose smooth functions g j > 0 g_j>0 g j > 0 B R ‾ \overline{B_R} B R g g g [14 ] provides smooth strictly plurisubharmonic solutions v j v_j v j B R ‾ \overline{B_R} B R g j n g_j^n g j n η \eta η ∣ z ∣ 2 − R 2 \lvert z\rvert^2-R^2 ∣ z ∣ 2 − R 2 η \eta η
∥ v j − v k ∥ L ∞ ( B R ) ≤ R 2 ∥ g j − g k ∥ L ∞ ( B R ) . (72) \lVert v_j-v_k\rVert_{L^\infty(B_R)}\leq R^2\lVert g_j-g_k\rVert_{L^\infty(B_R)}. \tag*{(72)} ∥ v j − v k ∥ L ∞ ( B R ) ≤ R 2 ∥ g j − g k ∥ L ∞ ( B R ) . (72) Since g j → g g_j\to g g j → g v j v_j v j v v v η \eta η
To identify the Monge–Ampère measure using decreasing convergence, use (4.6) to choose a subsequence with
∥ v j k + 1 − v j k ∥ ∞ ≤ 2 − k . (73) \lVert v_{j_{k+1}}-v_{j_k}\rVert_\infty\leq2^{-k}. \tag*{(73)} ∥ v j k + 1 − v j k ∥ ∞ ≤ 2 − k . (73) Define
c k = ∑ ℓ = k ∞ 2 − ℓ , v ~ k = v j k + c k . (74) c_k=\sum_{\ell=k}^{\infty}2^{-\ell},\qquad\widetilde{v}_k=v_{j_k}+c_k. \tag*{(74)} c k = ℓ = k ∑ ∞ 2 − ℓ , v k = v j k + c k . (74) By (73) and c k − c k + 1 = 2 − k c_k-c_{k+1}=2^{-k} c k − c k + 1 = 2 − k v ~ k + 1 ≤ v ~ k \widetilde v_{k+1}\leq\widetilde v_k v k + 1 ≤ v k v j k → v v_{j_k}\to v v j k → v c k → 0 c_k\to0 c k → 0 v ~ k ↓ v \widetilde v_k\downarrow v v k ↓ v c k c_k c k v j k v_{j_k} v j k g j k g_{j_k} g j k [5 ] yield
( − 1 ∂ ∂ ˉ v ) n = lim k → ∞ ( − 1 ∂ ∂ ˉ v ~ k ) n = lim k → ∞ g j k n ω 0 n = g n ω 0 n . (\sqrt{-1}\partial\bar\partial v)^n=\lim_{k\to\infty}(\sqrt{-1}\partial\bar\partial\widetilde v_k)^n=\lim_{k\to\infty}g_{j_k}^n\omega_0^n=g^n\omega_0^n. ( − 1 ∂ ∂ ˉ v ) n = k → ∞ lim ( − 1 ∂ ∂ ˉ v k ) n = k → ∞ lim g j k n ω 0 n = g n ω 0 n . Uniqueness follows from Lemma 4.1 with b = d = 0 b=d=0 b = d = 0
Approximate the potential without changing the estimate
We state the local form needed in the next section. We solve the approximating Dirichlet problems on B R B_R B R R = 3 L / 2 R=3L/2 R = 3 L /2 f f f B 2 L B_{2L} B 2 L f 1 / n f^{1/n} f 1/ n B R ‾ \overline{B_R} B R B L + 1 B_{L+1} B L + 1
Proposition 4.3 (The pseudo-estimate for continuous potentials). Fix 0 ≤ α ≤ 1 0\leq\alpha\leq1 0 ≤ α ≤ 1 0 < c ≤ 1 0<c\leq1 0 < c ≤ 1 C ≥ 1 C\geq1 C ≥ 1 f max > 0 f_{\max}>0 f m a x > 0 u ∈ C ( B R ‾ ) u\in C(\overline{B_R}) u ∈ C ( B R ) B R B_R B R R = 3 L / 2 R=3L/2 R = 3 L /2
( − 1 ∂ ∂ ˉ u ) n = f ω 0 n in B R , (\sqrt{-1}\partial\bar\partial u)^n=f\omega_0^n\quad\text{in }B_R, ( − 1 ∂ ∂ ˉ u ) n = f ω 0 n in B R , c ρ ψ α − 2 ≤ u ≤ C ρ ψ 1 − α + 2 on B L + 1 ‾ . (75) \frac{c\rho}{\psi^\alpha}-2\leq u\leq C\rho\psi^{1-\alpha}+2\quad\text{on }\overline{B_{L+1}}. \tag*{(75)} ψ α c ρ − 2 ≤ u ≤ C ρ ψ 1 − α + 2 on B L + 1 . (75) Assume that f ∈ C ( B 2 L ) f\in C(B_{2L}) f ∈ C ( B 2 L )
f ≤ f max , Δ R ( f 1 / n ) ≥ − K in B 2 L , (76) f\leq f_{\max},\qquad\Delta_{\mathbb R}(f^{1/n})\geq-K\quad\text{in }B_{2L}, \tag*{(76)} f ≤ f m a x , Δ R ( f 1/ n ) ≥ − K in B 2 L , (76) where K ≥ 0 K\geq0 K ≥ 0 2 C 2C 2 C
f ‾ max = ( f max 1 / n + 1 ) n . (77) \overline f_{\max}=(f_{\max}^{1/n}+1)^n. \tag*{(77)} f m a x = ( f m a x 1/ n + 1 ) n . (77) Use the definitions of V V V ε ± \varepsilon_\pm ε ± (31) , and of λ L \lambda_L λ L q L q_L q L (32) . For every fixed r ≥ 2 r\geq2 r ≥ 2 L r L_r L r L ≥ L r L\geq L_r L ≥ L r
0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (78) 0\leq G_tu(z)-u(z)\leq2t^2V(z)\varepsilon_-+(\lambda_Lt^2+Eq_L)V(z)\qquad(z\in B_r,\ 0<t\leq1). \tag*{(78)} 0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (78) Here E E E L L L f ≥ c 0 > 0 f\geq c_0>0 f ≥ c 0 > 0 B 2 L B_{2L} B 2 L c 0 / 2 c_0/2 c 0 /2
0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε + + E + q L V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (79) 0\leq G_tu(z)-u(z)\leq2t^2V(z)\varepsilon_++E+q_LV(z)\qquad(z\in B_r,\ 0<t\leq1). \tag*{(79)} 0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε + + E + q L V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (79) The constants in this latter estimate are independent of L L L c 0 c_0 c 0 K K K
Proof. We approximate the density root so that its Laplacian lower bound is unchanged. Write g = f 1 / n g=f^{1/n} g = f 1/ n χ \chi χ B 1 B_1 B 1 χ δ ( z ) = δ − 2 n χ ( z / δ ) \chi_\delta(z)=\delta^{-2n}\chi(z/\delta) χ δ ( z ) = δ − 2 n χ ( z / δ ) δ j ↓ 0 \delta_j\downarrow0 δ j ↓ 0 s j ↓ 0 s_j\downarrow0 s j ↓ 0 0 < δ j < L / 4 0<\delta_j<L/4 0 < δ j < L /4 0 < s j ≤ 1 0<s_j\leq1 0 < s j ≤ 1
g j = g ∗ χ δ j + s j , f j = g j n . (80) g_j=g*\chi_{\delta_j}+s_j,\qquad f_j=g_j^n. \tag*{(80)} g j = g ∗ χ δ j + s j , f j = g j n . (80) These functions are smooth on a neighborhood of B R ‾ \overline{B_R} B R (80) , the nonnegative kernel of mass one preserves the density bounds (76) . Together with (77) , these bounds give
g j > 0 , f j ≤ f ‾ max , Δ R g j = ( Δ R g ) ∗ χ δ j ≥ − K , ∥ g j − g ∥ L ∞ ( B R ) ⟶ 0. (81) g_j>0,\qquad f_j\leq\overline f_{\max},\qquad\Delta_{\mathbb R}g_j=(\Delta_{\mathbb R}g)*\chi_{\delta_j}\geq-K,\qquad\lVert g_j-g\rVert_{L^\infty(B_R)}\longrightarrow0. \tag*{(81)} g j > 0 , f j ≤ f m a x , Δ R g j = ( Δ R g ) ∗ χ δ j ≥ − K , ∥ g j − g ∥ L ∞ ( B R ) ⟶ 0. (81) The final convergence follows from uniform continuity of g g g B R ‾ \overline{B_R} B R f ≥ c 0 f \ge c_0 f ≥ c 0 f j ≥ c 0 / 2 f_j \ge c_0/2 f j ≥ c 0 /2 j j j
We next approximate the boundary values. Choose smooth functions η j \eta_j η j ∂ B R \partial B_R ∂ B R u ∣ ∂ B R u|_{\partial B_R} u ∣ ∂ B R u u u B 2 L B_{2L} B 2 L η j = ( u ∗ χ δ j ) ∣ ∂ B R \eta_j = (u * \chi_{\delta_j})|_{\partial B_R} η j = ( u ∗ χ δ j ) ∣ ∂ B R
The smooth Dirichlet theorem [14 ] gives a smooth strictly plurisubharmonic function w j w_j w j B R ‾ \overline{B_R} B R f j f_j f j η j \eta_j η j u u u
e j = ∥ η j − u ∥ L ∞ ( ∂ B R ) + R 2 ∥ g j − g ∥ L ∞ ( B R ) . (82) e_j = \|\eta_j-u\|_{L^\infty(\partial B_R)} + R^2\|g_j-g\|_{L^\infty(B_R)}. \tag*{(82)} e j = ∥ η j − u ∥ L ∞ ( ∂ B R ) + R 2 ∥ g j − g ∥ L ∞ ( B R ) . (82) Lemma 4.1 and (4.15) imply
∥ w j − u ∥ L ∞ ( B R ) ≤ e j ⟶ 0. (83) \|w_j-u\|_{L^\infty(B_R)} \le e_j \longrightarrow0. \tag*{(83)} ∥ w j − u ∥ L ∞ ( B R ) ≤ e j ⟶ 0. (83) Adding e j e_j e j w j w_j w j j j j
c ρ ψ 1 − α − 2 ≤ w j + e j ≤ C ρ ψ 1 − α + 2 + 2 e j ≤ 2 C ρ ψ 1 − α + 2 on B L + 1 ‾ . (84) \frac{c\rho}{\psi^{1-\alpha}}-2 \le w_j+e_j \le C\rho\psi^{1-\alpha}+2+2e_j \le2C\rho\psi^{1-\alpha}+2 \quad\text{on } \overline{B_{L+1}}. \tag*{(84)} ψ 1 − α c ρ − 2 ≤ w j + e j ≤ C ρ ψ 1 − α + 2 + 2 e j ≤ 2 C ρ ψ 1 − α + 2 on B L + 1 . (84) The last inequality uses e j ≤ 1 e_j\le1 e j ≤ 1 C ≥ 1 C\ge1 C ≥ 1 ρ ψ 1 − α ≥ e \rho\psi^{1-\alpha}\ge e ρ ψ 1 − α ≥ e
V j = w j + e j + J − a ρ A h ( s ) , V = u + J − a ρ A h ( s ) . (85) V_j=w_j+e_j+J-a\rho A_h(s), \qquad V=u+J-a\rho A_h(s). \tag*{(85)} V j = w j + e j + J − a ρ A h ( s ) , V = u + J − a ρ A h ( s ) . (85) By (4.17), V j → V V_j\to V V j → V B L + 1 ‾ \overline{B_{L+1}} B L + 1
Fix r r r L ≥ L r L\ge L_r L ≥ L r 0 < t ≤ 1 0<t\le1 0 < t ≤ 1 w j + e j w_j+e_j w j + e j
0 ≤ G t w j ( z ) − w j ( z ) ≤ 2 t 2 V j ( z ) ε − + ( λ L t 2 + E q L ) V j ( z ) ( z ∈ B r ) . (86) 0\le\mathcal{G}_t w_j(z)-w_j(z)\le2t^2V_j(z)\varepsilon^{-}+\left(\lambda_Lt^2+E_{qL}\right)V_j(z) \quad(z\in B_r). \tag*{(86)} 0 ≤ G t w j ( z ) − w j ( z ) ≤ 2 t 2 V j ( z ) ε − + ( λ L t 2 + E q L ) V j ( z ) ( z ∈ B r ) . (86) All parameters in this inequality were fixed before j j j G t w j → G t u \mathcal{G}_t w_j\to\mathcal{G}_t u G t w j → G t u B r B_r B r
If f ≥ c 0 f\ge c_0 f ≥ c 0 f f f
For a continuous potential on B 2 L B_{2L} B 2 L B 3 L / 2 B_{3L/2} B 3 L /2
Locally bounded Bedford–Taylor solutions
We now remove continuity of the potential. Let u u u B 2 L B_{2L} B 2 L L L L
( − 1 ∂ ∂ ˉ u ) n = f ω 0 n , c ρ ψ 1 − α ≤ u ≤ C ρ ψ 1 − α , (87) \left(\sqrt{-1}\partial\bar\partial u\right)^n=f\omega_0^n,\qquad\frac{c\rho}{\psi^{1-\alpha}}\le u\le C\rho\psi^{1-\alpha}, \tag*{(87)} ( − 1 ∂ ∂ ˉ u ) n = f ω 0 n , ψ 1 − α c ρ ≤ u ≤ C ρ ψ 1 − α , (87) with f ≥ 0 f\ge0 f ≥ 0 g = f 1 / n g=f^{1/n} g = f 1/ n u u u
Construct decreasing smooth boundary data
Set R = 3 L / 2 R=3L/2 R = 3 L /2 R ′ = 7 L / 4 R'=7L/4 R ′ = 7 L /4 B 2 L B_{2L} B 2 L B R B_R B R ϑ \vartheta ϑ B 1 B_1 B 1 ε j ↓ 0 \varepsilon_j \downarrow0 ε j ↓ 0 ε 1 < L / 8 \varepsilon_1<L/8 ε 1 < L /8
u j = u ∗ ϑ ε j , g j = g ∗ ϑ ε j . (88) u_j=u\ast\vartheta_{\varepsilon_j},\qquad g_j=g\ast\vartheta_{\varepsilon_j}. \tag*{(88)} u j = u ∗ ϑ ε j , g j = g ∗ ϑ ε j . (88) Both convolutions are defined on a neighborhood of B R ′ ‾ \overline{B_{R'}} B R ′ u j u_j u j u j ( z ) u_j(z) u j ( z ) u u u z z z u u u (88) , this monotonicity, the submean inequality, and upper semicontinuity give
u j ↓ u pointwise on B R ′ ‾ . (89) u_j\downarrow u\quad\text{pointwise on }\overline{B_{R'}}. \tag*{(89)} u j ↓ u pointwise on B R ′ . (89) Continuity of g g g g j → g g_j\to g g j → g B R ′ ‾ \overline{B_{R'}} B R ′
We next compare the n n n ( det D C 2 u j ) 1 / n (\det D_{\mathbb{C}}^2u_j)^{1/n} ( det D C 2 u j ) 1/ n g j g_j g j H H H det H = 1 \det H=1 det H = 1 q H ( z ) = H [ z ] q_H(z)=H[z] q H ( z ) = H [ z ] (87) gives
( − 1 ∂ ∂ ‾ u ) n = g n ω 0 n , \left(\sqrt{-1}\partial\overline{\partial}u\right)^n=g^n\omega_0^n, ( − 1 ∂ ∂ u ) n = g n ω 0 n , while det H = 1 \det H=1 det H = 1
( − 1 ∂ ∂ ‾ q H ) n = ω 0 n . \left(\sqrt{-1}\partial\overline{\partial}q_H\right)^n=\omega_0^n. ( − 1 ∂ ∂ q H ) n = ω 0 n . Dinew’s mixed Monge–Ampère inequality [22 Theorem 1.2] , applied on relatively compact balls to these two measures, gives
− 1 ∂ ∂ ‾ u ∧ ( − 1 ∂ ∂ ‾ q H ) n − 1 ≥ g ω 0 n . \sqrt{-1}\partial\overline{\partial}u\wedge\left(\sqrt{-1}\partial\overline{\partial}q_H\right)^{n-1}\ge g\omega_0^n. − 1 ∂ ∂ u ∧ ( − 1 ∂ ∂ q H ) n − 1 ≥ g ω 0 n . For n = 1 n=1 n = 1 H H H tr ( H − 1 ( u i j ˉ ) ) ≥ n g \operatorname{tr}(H^{-1}(u_{i\bar j}))\ge ng tr ( H − 1 ( u i j ˉ )) ≥ n g (88) gives
tr ( H − 1 D C 2 u j ) ≥ n g j . (90) \operatorname{tr}(H^{-1}D_{\mathbb{C}}^2u_j)\ge ng_j. \tag*{(90)} tr ( H − 1 D C 2 u j ) ≥ n g j . (90) For every positive semidefinite Hermitian matrix A A A
( det A ) 1 / n = inf H > 0 det H = 1 1 n tr ( H − 1 A ) . (91) (\det A)^{1/n}=\inf_{\substack{H>0\\ \det H=1}}\frac{1}{n}\operatorname{tr}(H^{-1}A). \tag*{(91)} ( det A ) 1/ n = H > 0 d e t H = 1 inf n 1 tr ( H − 1 A ) . (91) For A > 0 A>0 A > 0 H = ( det A ) − 1 / n A H=(\det A)^{-1/n}A H = ( det A ) − 1/ n A A A A (91) to (90) , which holds for every H > 0 H>0 H > 0 det H = 1 \det H=1 det H = 1
( det D C 2 u j ) 1 / n ≥ g j . (92) (\det D_{\mathbb{C}}^2u_j)^{1/n}\ge g_j. \tag*{(92)} ( det D C 2 u j ) 1/ n ≥ g j . (92) The root g j g_j g j g g g a j ↓ 0 a_j\downarrow0 a j ↓ 0
a j ≥ ∥ g j − g ∥ L ∞ ( B R ′ ‾ ) , ϕ j = u j + a j ρ . (93) a_j\ge\lVert g_j-g\rVert_{L^\infty(\overline{B_{R'}})},\qquad\phi_j=u_j+a_j\rho. \tag*{(93)} a j ≥ ∥ g j − g ∥ L ∞ ( B R ′ ) , ϕ j = u j + a j ρ . (93) For example, take the supremum of the norms in (93) over all indices at least j j j (67) and (92) give
( det D C 2 ϕ j ) 1 / n ≥ g j + a j ≥ g , ϕ j ↓ u . (94) (\det D_{\mathbb{C}}^2\phi_j)^{1/n}\ge g_j+a_j\ge g,\qquad\phi_j\downarrow u. \tag*{(94)} ( det D C 2 ϕ j ) 1/ n ≥ g j + a j ≥ g , ϕ j ↓ u . (94) The second assertion follows from (89) and (93) . Thus (94) supplies smooth subsolutions ϕ j \phi_j ϕ j ∂ B R \partial B_R ∂ B R
Solve the exact equation and identify the limit
Lemma 4.2 supplies continuous plurisubharmonic solutions v j v_j v j B R ‾ \overline{B_R} B R
( − 1 ∂ ∂ ‾ v j ) n = f ω 0 n in B R , v j = ϕ j on ∂ B R . (95) (\sqrt{-1}\partial\overline{\partial}v_j)^n=f\omega_0^n\quad\text{in }B_R,\qquad v_j=\phi_j\quad\text{on }\partial B_R. \tag*{(95)} ( − 1 ∂ ∂ v j ) n = f ω 0 n in B R , v j = ϕ j on ∂ B R . (95) The Bedford–Taylor comparison principle [5 ] , applied to (95) and (94) , gives v j ≥ ϕ j v_j\geq\phi_j v j ≥ ϕ j
v j + 1 ≤ v j , v j ≥ ϕ j ≥ u on B R . (96) v_{j+1}\leq v_j,\qquad v_j\geq\phi_j\geq u\quad\text{on }B_R. \tag*{(96)} v j + 1 ≤ v j , v j ≥ ϕ j ≥ u on B R . (96) By (96) , v j v_j v j u u u u u u B R ′ B_{R'} B R ′ u u u
v ~ j ( z ) = { v j ( z ) , z ∈ B R , ϕ j ( z ) , z ∈ B R ′ ∖ B R . (97) \widetilde v_j(z)=
\begin{cases}
v_j(z), & z\in B_R,\\
\phi_j(z), & z\in B_{R'}\setminus B_R.
\end{cases} \tag*{(97)} v j ( z ) = { v j ( z ) , ϕ j ( z ) , z ∈ B R , z ∈ B R ′ ∖ B R . (97) The two definitions agree on ∂ B R \partial B_R ∂ B R (96) , v j ≥ ϕ j v_j\geq\phi_j v j ≥ ϕ j max ( v j , ϕ j ) \max(v_j,\phi_j) max ( v j , ϕ j ) ϕ j \phi_j ϕ j v ~ j \widetilde v_j v j B R ′ B_{R'} B R ′
The definition (97) and (95) give Monge–Ampère measure f ω 0 n f\omega_0^n f ω 0 n B R B_R B R B R B_R B R (94) . The remaining sphere has zero ω 0 n \omega_0^n ω 0 n
( − 1 ∂ ∂ ‾ v ~ j ) n ≥ f ω 0 n on B R ′ . (98) (\sqrt{-1}\partial\overline{\partial}\widetilde v_j)^n\geq f\omega_0^n\quad\text{on }B_{R'}. \tag*{(98)} ( − 1 ∂ ∂ v j ) n ≥ f ω 0 n on B R ′ . (98) By (96) inside B R B_R B R (94) outside, the functions v ~ j \widetilde v_j v j u u u v ~ 1 \widetilde v_1 v 1 v ~ \widetilde v v [5 ] and (98) give, using (87) ,
( − 1 ∂ ∂ ‾ v ~ ) n ≥ f ω 0 n = ( − 1 ∂ ∂ ‾ u ) n , v ~ ≥ u . (99) (\sqrt{-1}\partial\overline{\partial}\widetilde v)^n\geq f\omega_0^n=(\sqrt{-1}\partial\overline{\partial}u)^n,\qquad\widetilde v\geq u. \tag*{(99)} ( − 1 ∂ ∂ v ) n ≥ f ω 0 n = ( − 1 ∂ ∂ u ) n , v ≥ u . (99) Moreover, (94) gives v ~ = u \widetilde v=u v = u B R ′ ∖ B R B_{R'}\setminus B_R B R ′ ∖ B R
Choose R ^ \widehat R R R < R ^ < R ′ R<\widehat R<R' R < R < R ′ ∂ B R ^ \partial B_{\widehat R} ∂ B R u u u [27 Lemma 1.6] to this boundary equality and (99) gives v ~ ≤ u \widetilde v\leq u v ≤ u B R ^ B_{\widehat R} B R (99) also gives v ~ ≥ u \widetilde v\geq u v ≥ u
v j ↓ u pointwise on B R . (100) v_j\downarrow u\quad\text{pointwise on }B_R. \tag*{(100)} v j ↓ u pointwise on B R . (100) Pass the local pseudo-estimate to the limit
The lower bound in (87) and the ordering (96) give v j ≥ c ρ / ψ α v_j\geq c\rho/\psi^\alpha v j ≥ c ρ / ψ α 0 < η ≤ 1 0<\eta\leq1 0 < η ≤ 1 L + 1 < R L+1<R L + 1 < R B L + 1 ‾ \overline{B_{L+1}} B L + 1 B R B_R B R j j j
c ρ ψ α ≤ v j ≤ C ρ ψ 1 − α + η ≤ C ρ ψ 1 − α + 2 on B L + 1 ‾ . (101) \frac{c\rho}{\psi^\alpha}\leq v_j\leq C\rho\psi^{1-\alpha}+\eta\leq C\rho\psi^{1-\alpha}+2\quad\text{on }\overline{B_{L+1}}. \tag*{(101)} ψ α c ρ ≤ v j ≤ C ρ ψ 1 − α + η ≤ C ρ ψ 1 − α + 2 on B L + 1 . (101) Indeed, if no such index existed, the sets
E j = { z ∈ B L + 1 ‾ : v j ( z ) − C ρ ( z ) ψ ( z ) 1 − α ≥ η } E_j=\left\{z\in\overline{B_{L+1}}:v_j(z)-C\rho(z)\psi(z)^{1-\alpha}\geq\eta\right\} E j = { z ∈ B L + 1 : v j ( z ) − C ρ ( z ) ψ ( z ) 1 − α ≥ η } would be nonempty. By (96) and continuity of v j v_j v j u − C ρ ψ 1 − α ≥ η u-C\rho\psi^{1-\alpha}\geq\eta u − C ρ ψ 1 − α ≥ η (100) , contradicting (87) . This argument uses continuity of the replacements, not of u u u
Apply Proposition 4.3 to v j v_j v j (101) . Its fixed numerical offsets are already covered by that proposition. Choose the buffered data (77) and all normalization parameters once, independently of j j j L L L
V j = v j + J − a ρ A h ( s ) , V = u + J − a ρ A h ( s ) . V_j=v_j+J-a\rho A_h(s),\qquad V=u+J-a\rho A_h(s). V j = v j + J − a ρ A h ( s ) , V = u + J − a ρ A h ( s ) . Equation (100) gives V j ↓ V V_j\downarrow V V j ↓ V (36) gives V > 0 V>0 V > 0 r ≥ 2 r\geq2 r ≥ 2 L ≥ L r L\geq L_r L ≥ L r 0 < t ≤ 1 0<t\leq1 0 < t ≤ 1 (78) gives
0 ≤ G t v j ( z ) − v j ( z ) ≤ 2 t 2 V j ( z ) ε − + ( λ L t 2 + E q L ) V j ( z ) ( z ∈ B r ) . (102) 0\leq\mathcal{G}_t v_j(z)-v_j(z)\leq2t^2V_j(z)^{\varepsilon_-}+(\lambda_Lt^2+E_{q_L})V_j(z)\qquad(z\in B_r). \tag*{(102)} 0 ≤ G t v j ( z ) − v j ( z ) ≤ 2 t 2 V j ( z ) ε − + ( λ L t 2 + E q L ) V j ( z ) ( z ∈ B r ) . (102) The constants in this inequality do not depend on j j j f ≥ c 0 > 0 f\geq c_0>0 f ≥ c 0 > 0 (79) gives
0 ≤ G t v j ( z ) − v j ( z ) ≤ 2 t 2 V j ( z ) ε + + E + q L V j ( z ) . (103) 0\leq\mathcal{G}_t v_j(z)-v_j(z)\leq2t^2V_j(z)^{\varepsilon_+}+E_{+q_L}V_j(z). \tag*{(103)} 0 ≤ G t v j ( z ) − v j ( z ) ≤ 2 t 2 V j ( z ) ε + + E + q L V j ( z ) . (103) For fixed z ∈ B r z\in B_r z ∈ B r 0 < t ≤ 1 0<t\leq1 0 < t ≤ 1 z + B ‾ t z+\overline{B}_t z + B t B R B_R B R L ≥ L r L\geq L_r L ≥ L r (100) gives pointwise convergence, and the functions are bounded between u u u v 1 v_1 v 1 (8) has mass one, so these bounds and pointwise convergence permit bounded convergence:
G t v j ( z ) ⟶ G t u ( z ) , v j ( z ) ⟶ u ( z ) . \mathcal{G}_t v_j(z)\longrightarrow\mathcal{G}_t u(z),\qquad v_j(z)\longrightarrow u(z). G t v j ( z ) ⟶ G t u ( z ) , v j ( z ) ⟶ u ( z ) . Thus the difference also converges; no monotonicity of G t v j − v j \mathcal{G}_t v_j-v_j G t v j − v j j → ∞ j\to\infty j → ∞ (102) proves
0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (104) 0\leq\mathcal{G}_t u(z)-u(z)\leq2t^2V(z)^{\varepsilon_-}+(\lambda_Lt^2+E_{q_L})V(z)\qquad(z\in B_r,\ 0<t\leq1). \tag*{(104)} 0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε − + ( λ L t 2 + E q L ) V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (104) If f ≥ c 0 f\geq c_0 f ≥ c 0 (103) proves
0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε + + E + q L V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (105) 0\leq\mathcal{G}_t u(z)-u(z)\leq2t^2V(z)^{\varepsilon_+}+E_{+q_L}V(z)\qquad(z\in B_r,\ 0<t\leq1). \tag*{(105)} 0 ≤ G t u ( z ) − u ( z ) ≤ 2 t 2 V ( z ) ε + + E + q L V ( z ) ( z ∈ B r , 0 < t ≤ 1 ) . (105) This completes the proof of Theorem 1.5 . The approximation limit is taken at each fixed L L L K = 0 K=0 K = 0 λ L = 0 \lambda_L=0 λ L = 0 (32) . After taking the approximation limit, we let L → ∞ L\to\infty L → ∞ q L → 0 q_L\to0 q L → 0
From the pseudo-estimate to a real C 2 C^2 C 2
At a fixed finite radius, the additive term in (12) survives as t ↓ 0 t\downarrow0 t ↓ 0 C 2 C^2 C 2
Constant-density perturbations
We first use Savin’s theorem to control a constant-density solution that is close to a quadratic polynomial with positive definite complex Hessian. The resulting estimate controls real derivatives. For a real symmetric matrix N N N H ( N ) \mathcal{H}(N) H ( N ) H ( D C 2 v ) = D C 2 v \mathcal{H}(D_{\mathbb{C}}^2v)=D_{\mathbb{C}}^2v H ( D C 2 v ) = D C 2 v N ≥ 0 N\geq0 N ≥ 0
H ( N ) ≥ 0 , tr H ( N ) = 1 4 tr N . (106) \mathcal{H}(N)\geq0,\qquad\operatorname{tr}\mathcal{H}(N)=\frac{1}{4}\operatorname{tr}N. \tag*{(106)} H ( N ) ≥ 0 , tr H ( N ) = 4 1 tr N . (106) Lemma 6.1 (Constant-density perturbations). Fix 0 < λ ≤ Λ 0 < \lambda\leq\Lambda 0 < λ ≤ Λ β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 ) e ∗ > 0 e_\ast> 0 e ∗ > 0 C C C n , λ , Λ n,\lambda,\Lambda n , λ , Λ P P P H = D C 2 P H = D_{\mathbb{C}}^2P H = D C 2 P λ I ≤ H ≤ Λ I \lambda I \leq H \leq\Lambda I λ I ≤ H ≤ Λ I v ∈ C ( B r ) ∩ PSH ( B r ) v \in C(B_r) \cap\operatorname{PSH}(B_r) v ∈ C ( B r ) ∩ PSH ( B r )
det D C 2 v = det H , ∥ v − P ∥ L ∞ ( B r ) ≤ e r 2 , 0 ≤ e ≤ e ∗ . (107) \det D_{\mathbb{C}}^2v = \det H,\qquad\lVert v-P\rVert_{L^\infty(B_r)} \leq er^2,\qquad0 \leq e \leq e_\ast. \tag*{(107)} det D C 2 v = det H , ∥ v − P ∥ L ∞ ( B r ) ≤ e r 2 , 0 ≤ e ≤ e ∗ . (107) The equation is understood in the Bedford–Taylor sense. Then v ∈ C 2 , β ( B r / 2 ) v \in C^{2,\beta}(B_{r/2}) v ∈ C 2 , β ( B r /2 )
r − 2 ∥ v − P ∥ L ∞ ( B r / 2 ) + r − 1 ∥ D R ( v − P ) ∥ L ∞ ( B r / 2 ) + ∥ D R 2 ( v − P ) ∥ L ∞ ( B r / 2 ) + r β [ D R 2 ( v − P ) ] C β ( B r / 2 ) ≤ C e . (108) r^{-2}\lVert v-P\rVert_{L^\infty(B_{r/2})}+r^{-1}\lVert D_{\mathbb{R}}(v-P)\rVert_{L^\infty(B_{r/2})}
+\lVert D_{\mathbb{R}}^2(v-P)\rVert_{L^\infty(B_{r/2})}
+r^\beta[D_{\mathbb{R}}^2(v-P)]_{C^\beta(B_{r/2})}\leq Ce. \tag*{(108)} r − 2 ∥ v − P ∥ L ∞ ( B r /2 ) + r − 1 ∥ D R ( v − P ) ∥ L ∞ ( B r /2 ) + ∥ D R 2 ( v − P ) ∥ L ∞ ( B r /2 ) + r β [ D R 2 ( v − P ) ] C β ( B r /2 ) ≤ C e . (108) Proof. Rescale by z = r y z=ry z = r y r 2 r^2 r 2 r = 1 r=1 r = 1 w = v − P w=v-P w = v − P c = det H c=\det H c = det H
F H ( N ) = { det ( H + H ( N ) ) 1 / n − c 1 / n , H + H ( N ) ≥ 0 , − c 1 / n , H + H ( N ) ≱ 0. (109) F_H(N)=
\begin{cases}
\det\bigl(H+\mathcal{H}(N)\bigr)^{1/n}-c^{1/n}, & H+\mathcal{H}(N)\geq0,\\
-c^{1/n}, & H+\mathcal{H}(N)\not\geq0.
\end{cases} \tag*{(109)} F H ( N ) = { det ( H + H ( N ) ) 1/ n − c 1/ n , − c 1/ n , H + H ( N ) ≥ 0 , H + H ( N ) ≥ 0. (109) The operator in (6.4) is continuous and degenerate elliptic on all real symmetric matrices. It is smooth near zero and satisfies F H ( 0 ) = 0 F_H(0)=0 F H ( 0 ) = 0
D F H ( 0 ) [ N ] = c 1 / n n tr ( H − 1 H ( N ) ) . DF_H(0)[N]=\frac{c^{1/n}}{n}\operatorname{tr}\bigl(H^{-1}\mathcal{H}(N)\bigr). D F H ( 0 ) [ N ] = n c 1/ n tr ( H − 1 H ( N ) ) . The Hessian bounds in Lemma 6.1 and (6.1) give uniform real ellipticity in a fixed neighborhood of zero. The size of this neighborhood and the bounds for the first two derivatives of F H F_H F H H H H
By the equivalence between continuous pluripotential and viscosity solutions [27 Proposition 1.5 and Lemma 4.7(1)] , the equation for v v v C 2 C^2 C 2 φ \varphi φ w = v − P w=v-P w = v − P P + φ P+\varphi P + φ v v v D C 2 ( P + φ ) = H + H ( D R 2 φ ) D_{\mathbb{C}}^2(P+\varphi)=H+\mathcal{H}(D_{\mathbb{R}}^2\varphi) D C 2 ( P + φ ) = H + H ( D R 2 φ ) v v v H + H ( D R 2 φ ) ≥ 0 H+\mathcal{H}(D_{\mathbb{R}}^2\varphi)\geq0 H + H ( D R 2 φ ) ≥ 0 v v v F H ( D R 2 w ) = 0 F_H(D_{\mathbb{R}}^2w)=0 F H ( D R 2 w ) = 0 F H F_H F H [45 Theorem 1.3] : global degenerate ellipticity and smooth uniform ellipticity near the zero solution suffice.
Choose the ellipticity neighborhood for F H F_H F H C 2 , β C^{2,\beta} C 2 , β λ / 2 \lambda/2 λ /2 H H H [45 Theorem 1.3] on interior balls covering B 3 / 4 B_{3/4} B 3/4 e ∗ e_\ast e ∗ C 2 , β C^{2,\beta} C 2 , β ∥ D C 2 w ∥ L ∞ ( B 3 / 4 ) ≤ λ / 2 \lVert D_{\mathbb{C}}^2w\rVert_{L^\infty(B_{3/4})}\leq\lambda/2 ∥ D C 2 w ∥ L ∞ ( B 3/4 ) ≤ λ /2 det D C 2 v = det H \det D_{\mathbb{C}}^2v=\det H det D C 2 v = det H v = P + w v=P+w v = P + w
0 = log det ( H + D C 2 w ) − log det H = ∫ 0 1 tr ( ( H + s D C 2 w ) − 1 D C 2 w ) d s . 0=\log\det\bigl(H+D_{\mathbb{C}}^2w\bigr)-\log\det H
=\int_0^1\operatorname{tr}\bigl((H+sD_{\mathbb{C}}^2w)^{-1}D_{\mathbb{C}}^2w\bigr)\,ds. 0 = log det ( H + D C 2 w ) − log det H = ∫ 0 1 tr ( ( H + s D C 2 w ) − 1 D C 2 w ) d s . This is a linear uniformly elliptic equation for w w w C β C^\beta C β ∥ w ∥ L ∞ ( B 3 / 4 ) ≤ e \lVert w\rVert_{L^\infty(B_{3/4})}\leq e ∥ w ∥ L ∞ ( B 3/4 ) ≤ e
∥ w ∥ C 2 , β ( B 1 / 2 ) ≤ C ∥ w ∥ L ∞ ( B 3 / 4 ) ≤ C e . \lVert w\rVert_{C^{2,\beta}(B_{1/2})}\leq C\lVert w\rVert_{L^\infty(B_{3/4})}\leq Ce. ∥ w ∥ C 2 , β ( B 1/2 ) ≤ C ∥ w ∥ L ∞ ( B 3/4 ) ≤ C e . Scaling back proves (6.3).
To treat a variable density, we replace f f f u u u B r ( x ) ‾ \overline{B_r(x)} B r ( x ) B r ( x ) B_r(x) B r ( x )
( − 1 ∂ ∂ ‾ u ) n = f ω 0 n (\sqrt{-1}\partial\overline{\partial}u)^n=f\omega_0^n ( − 1 ∂ ∂ u ) n = f ω 0 n there in the Bedford–Taylor sense. Assume that f f f B r ( x ) ‾ \overline{B_r(x)} B r ( x ) 0 < c 0 ≤ f ≤ c 1 0<c_0\le f\le c_1 0 < c 0 ≤ f ≤ c 1 c = f ( x ) c=f(x) c = f ( x ) v v v
det D C 2 v = c in B r ( x ) , v = u on ∂ B r ( x ) . (110) \det D_{\mathbb{C}}^2v=c\quad\text{in }B_r(x),\qquad v=u\quad\text{on }\partial B_r(x). \tag*{(110)} det D C 2 v = c in B r ( x ) , v = u on ∂ B r ( x ) . (110) The continuous Dirichlet solution exists by the Bedford–Taylor theory [6 ] . If f f f ω \omega ω
∥ u − v ∥ L ∞ ( B r ( x ) ) ≤ r 2 ∥ f 1 / n − c 1 / n ∥ L ∞ ( B r ( x ) ) ≤ C r 2 ω ( r ) , (111) \|u-v\|_{L^\infty(B_r(x))}\le r^2\|f^{1/n}-c^{1/n}\|_{L^\infty(B_r(x))}\le Cr^2\omega(r), \tag*{(111)} ∥ u − v ∥ L ∞ ( B r ( x )) ≤ r 2 ∥ f 1/ n − c 1/ n ∥ L ∞ ( B r ( x )) ≤ C r 2 ω ( r ) , (111) where C C C n , c 0 , c 1 n,c_0,c_1 n , c 0 , c 1 P P P det D C 2 P = c \det D_{\mathbb{C}}^2P=c det D C 2 P = c ∥ u − P ∥ L ∞ ( B r ( x ) ) ≤ ϵ r 2 \|u-P\|_{L^\infty(B_r(x))}\le\epsilon r^2 ∥ u − P ∥ L ∞ ( B r ( x )) ≤ ϵ r 2 P ± ϵ r 2 P\pm\epsilon r^2 P ± ϵ r 2
∥ v − P ∥ L ∞ ( B r ( x ) ) ≤ ϵ r 2 . (112) \|v-P\|_{L^\infty(B_r(x))}\le\epsilon r^2. \tag*{(112)} ∥ v − P ∥ L ∞ ( B r ( x )) ≤ ϵ r 2 . (112) These estimates require no derivative bounds at the boundary of the comparison ball.
Summable quadratic approximations
We next combine (6.3) with (6.6). The Dini tail D ω \mathcal{D}_\omega D ω r ↓ 0 r\downarrow0 r ↓ 0
Lemma 6.2 (A local quadratic approximation criterion). Let u ∈ C ( B 3 / 2 ) ∩ PSH ( B 3 / 2 ) u\in C(B_{3/2})\cap\operatorname{PSH}(B_{3/2}) u ∈ C ( B 3/2 ) ∩ PSH ( B 3/2 ) det D C 2 u = f \det D_{\mathbb{C}}^2u=f det D C 2 u = f c 0 ≤ f ≤ c 1 c_0\le f\le c_1 c 0 ≤ f ≤ c 1 f f f ω \omega ω 0 < λ ≤ Λ 0<\lambda\le\Lambda 0 < λ ≤ Λ δ > 0 \delta>0 δ > 0 n , c 0 , c 1 , λ , Λ n,c_0,c_1,\lambda,\Lambda n , c 0 , c 1 , λ , Λ 0 < r 0 < 1 / 8 0<r_0<1/8 0 < r 0 < 1/8 x ∈ B 5 / 4 x\in B_{5/4} x ∈ B 5/4 P x P_x P x
λ I ≤ D C 2 P x ≤ Λ I , det D C 2 P x = f ( x ) , \lambda I\le D_{\mathbb{C}}^2P_x\le\Lambda I,\qquad\det D_{\mathbb{C}}^2P_x=f(x), λ I ≤ D C 2 P x ≤ Λ I , det D C 2 P x = f ( x ) , ∥ u − P x ∥ L ∞ ( B r 0 ( x ) ) ≤ e 0 r 0 2 , e 0 + D ω ( r 0 ) ≤ δ , (113) \|u-P_x\|_{L^\infty(B_{r_0}(x))}\le e_0r_0^2,\qquad e_0+\mathcal{D}_\omega(r_0)\le\delta, \tag*{(113)} ∥ u − P x ∥ L ∞ ( B r 0 ( x )) ≤ e 0 r 0 2 , e 0 + D ω ( r 0 ) ≤ δ , (113) and
sup x ∈ B 5 / 4 ∑ ℓ = 0 2 ∣ D R ℓ P x ( x ) ∣ ≤ J . (114) \sup_{x\in B_{5/4}}\sum_{\ell=0}^{2}|D_{\mathbb{R}}^\ell P_x(x)|\le J. \tag*{(114)} x ∈ B 5/4 sup ℓ = 0 ∑ 2 ∣ D R ℓ P x ( x ) ∣ ≤ J . (114) Then u ∈ C 2 ( B 5 / 4 ) u\in C^2(B_{5/4}) u ∈ C 2 ( B 5/4 )
∥ u ∥ C 2 ( B 5 / 4 ) ≤ C ( J + e 0 + D ω ( r 0 ) ) . (115) \|u\|_{C^2(B_{5/4})}\le C\bigl(J+e_0+\mathcal{D}_\omega(r_0)\bigr). \tag*{(115)} ∥ u ∥ C 2 ( B 5/4 ) ≤ C ( J + e 0 + D ω ( r 0 ) ) . (115) Its real Hessian has a common modulus of continuity determined by n , c 0 , c 1 , λ , Λ , r 0 , e 0 , ω n,c_0,c_1,\lambda,\Lambda,r_0,e_0,\omega n , c 0 , c 1 , λ , Λ , r 0 , e 0 , ω
Proof. Fix a centre x x x P 0 = P x P_0=P_x P 0 = P x c = f ( x ) c=f(x) c = f ( x ) λ 2 I ≤ H ≤ 2 Λ I \frac{\lambda}{2}I\le H\le2\Lambda I 2 λ I ≤ H ≤ 2Λ I P k P_k P k B r k B_{r_k} B r k r k = θ k r 0 r_k=\theta^kr_0 r k = θ k r 0 θ ∈ ( 0 , 1 / 4 ) \theta\in(0,1/4) θ ∈ ( 0 , 1/4 )
∥ u − P k ∥ L ∞ ( B r k ) ≤ e k r k 2 , det D C 2 P k = c . (116) \|u-P_k\|_{L^\infty(B_{r_k})}\le e_kr_k^2,\qquad\det D_{\mathbb{C}}^2P_k=c. \tag*{(116)} ∥ u − P k ∥ L ∞ ( B r k ) ≤ e k r k 2 , det D C 2 P k = c . (116) At the k k k
e k ≤ e ∗ , λ 2 I ≤ D C 2 P k ≤ 2 Λ I . (117) e_k\le e_*,\qquad\frac{\lambda}{2}I\le D_{\mathbb{C}}^2P_k\le2\Lambda I. \tag*{(117)} e k ≤ e ∗ , 2 λ I ≤ D C 2 P k ≤ 2Λ I . (117) We verify below that these conditions hold at every step. Let v k v_k v k B r k B_{r_k} B r k c c c u u u v k v_k v k ϵ k r k 2 \epsilon_k r_k^2 ϵ k r k 2 P k P_k P k v k ∈ C 2 , β ( B r k / 2 ) v_k \in C^{2,\beta}(B_{r_k/2}) v k ∈ C 2 , β ( B r k /2 )
∥ u − v k ∥ L ∞ ( B r k ) ≤ C r k 2 ω ( r k ) , r k β [ D R 2 ( v k − P k ) ] C β ( B r k / 2 ) ≤ C ϵ k . \lVert u-v_k\rVert_{L^\infty(B_{r_k})} \leq Cr_k^2\omega(r_k), \qquad r_k^\beta[D^2_{\mathbb{R}}(v_k-P_k)]_{C^\beta(B_{r_k/2})} \leq C\epsilon_k. ∥ u − v k ∥ L ∞ ( B r k ) ≤ C r k 2 ω ( r k ) , r k β [ D R 2 ( v k − P k ) ] C β ( B r k /2 ) ≤ C ϵ k . We may therefore define P k + 1 P_{k+1} P k + 1 v k v_k v k B θ r k B_{\theta r_k} B θ r k
( θ r k ) − 2 ∥ u − P k + 1 ∥ L ∞ ( B θ r k ) ≤ C θ β ϵ k + C θ − 2 ω ( r k ) , (\theta r_k)^{-2}\lVert u-P_{k+1}\rVert_{L^\infty(B_{\theta r_k})} \leq C\theta^\beta\epsilon_k+C\theta^{-2}\omega(r_k), ( θ r k ) − 2 ∥ u − P k + 1 ∥ L ∞ ( B θ r k ) ≤ C θ β ϵ k + C θ − 2 ω ( r k ) , ∣ D R ℓ ( P k + 1 − P k ) ( 0 ) ∣ ≤ C ϵ k r k 2 − ℓ , ℓ = 0 , 1 , 2. (118) \lvert D^\ell_{\mathbb{R}}(P_{k+1}-P_k)(0)\rvert\leq C\epsilon_k r_k^{2-\ell}, \qquad\ell=0,1,2. \tag*{(118)} ∣ D R ℓ ( P k + 1 − P k ) ( 0 )∣ ≤ C ϵ k r k 2 − ℓ , ℓ = 0 , 1 , 2. (118) Since v k v_k v k det D C 2 v k = c \det D^2_{\mathbb{C}}v_k=c det D C 2 v k = c P k + 1 P_{k+1} P k + 1 v k v_k v k det D C 2 P k + 1 = c \det D^2_{\mathbb{C}}P_{k+1}=c det D C 2 P k + 1 = c
Choose θ \theta θ C θ β ≤ 1 / 2 C\theta^\beta\leq1/2 C θ β ≤ 1/2
ϵ k + 1 = 1 2 ϵ k + C θ − 2 ω ( θ k r 0 ) . (119) \epsilon_{k+1}=\frac{1}{2}\epsilon_k+C\theta^{-2}\omega(\theta^k r_0). \tag*{(119)} ϵ k + 1 = 2 1 ϵ k + C θ − 2 ω ( θ k r 0 ) . (119) Summing (6.14) and using monotonicity of ω \omega ω
∑ k = 0 ∞ ϵ k ≤ C ( ϵ 0 + ∑ k = 0 ∞ ω ( θ k r 0 ) ) ≤ C ( ϵ 0 + D ω ( r 0 ) ) . (120) \begin{aligned}
\sum_{k=0}^{\infty}\epsilon_k\leq C\left(\epsilon_0+\sum_{k=0}^{\infty}\omega(\theta^k r_0)\right) \\
&\leq C(\epsilon_0+D_\omega(r_0)). \tag*{(120)}
\end{aligned} k = 0 ∑ ∞ ϵ k ≤ C ( ϵ 0 + k = 0 ∑ ∞ ω ( θ k r 0 ) ) ≤ C ( ϵ 0 + D ω ( r 0 )) . (120) Indeed, for k ≥ 1 k\geq1 k ≥ 1
∣ log θ ∣ ω ( θ k r 0 ) ≤ ∫ θ k r 0 θ k − 1 r 0 ω ( s ) s d s . \lvert\log\theta\rvert\omega(\theta^k r_0)\leq\int_{\theta^k r_0}^{\theta^{k-1}r_0}\frac{\omega(s)}{s}\,ds. ∣ log θ ∣ ω ( θ k r 0 ) ≤ ∫ θ k r 0 θ k − 1 r 0 s ω ( s ) d s . By the initial smallness condition (6.8), we can choose δ \delta δ ϵ k \epsilon_k ϵ k ϵ ∗ \epsilon_* ϵ ∗ min { λ / 2 , Λ } \min\{\lambda/2,\Lambda\} min { λ /2 , Λ }
The three coefficient estimates in (6.13) show that P k P_k P k P ∞ , x P_{\infty,x} P ∞ , x B r k ( x ) B_{r_k}(x) B r k ( x )
r k − 2 ∥ u − P ∞ , x ∥ L ∞ ( B r k ( x ) ) ≤ C ∑ j = k ∞ ϵ j ⟶ 0. (121) r_k^{-2}\lVert u-P_{\infty,x}\rVert_{L^\infty(B_{r_k}(x))}\leq C\sum_{j=k}^{\infty}\epsilon_j\longrightarrow0. \tag*{(121)} r k − 2 ∥ u − P ∞ , x ∥ L ∞ ( B r k ( x )) ≤ C j = k ∑ ∞ ϵ j ⟶ 0. (121) The sequence on the right is independent of x x x r k r_k r k θ − 2 \theta^{-2} θ − 2 η ( r ) → 0 \eta(r)\to0 η ( r ) → 0
∥ u − P ∞ , x ∥ L ∞ ( B r ( x ) ) ≤ η ( r ) r 2 ( x ∈ B 5 / 4 , 0 < r ≤ r 0 ) . (122) \lVert u-P_{\infty,x}\rVert_{L^\infty(B_r(x))}\leq\eta(r)r^2 \qquad(x\in B_{5/4},\ 0<r\leq r_0). \tag*{(122)} ∥ u − P ∞ , x ∥ L ∞ ( B r ( x )) ≤ η ( r ) r 2 ( x ∈ B 5/4 , 0 < r ≤ r 0 ) . (122) For completeness, this uniform expansion gives classical second derivatives, not only pointwise quadratic expansions. Write the limiting polynomial as
P ∞ , x ( z ) = a x + b x ⋅ ( z − x ) + 1 2 ( z − x ) T M x ( z − x ) . P_{\infty,x}(z)=a_x+b_x\cdot(z-x)+\frac{1}{2}(z-x)^T M_x(z-x). P ∞ , x ( z ) = a x + b x ⋅ ( z − x ) + 2 1 ( z − x ) T M x ( z − x ) . The expansion (6.17) gives a x = u ( x ) a_x=u(x) a x = u ( x ) D R u ( x ) = b x D_{\mathbb{R}}u(x)=b_x D R u ( x ) = b x 0 < d = ∣ x − y ∣ < r 0 / 4 0<d=\lvert x-y\rvert<r_0/4 0 < d = ∣ x − y ∣ < r 0 /4 d / 2 d/2 d /2 ( x + y ) / 2 (x+y)/2 ( x + y ) /2 4 η ( 2 d ) d 2 4\eta(2d)d^2 4 η ( 2 d ) d 2
∣ M x − M y ∣ ≤ C η ( 2 d ) , ∣ b y − b x − M x ( y − x ) ∣ ≤ C η ( 2 d ) d . (123) |M_x-M_y|\le C\eta(2d),\qquad|b_y-b_x-M_x(y-x)|\le C\eta(2d)d. \tag*{(123)} ∣ M x − M y ∣ ≤ C η ( 2 d ) , ∣ b y − b x − M x ( y − x ) ∣ ≤ C η ( 2 d ) d . (123) Since D R u ( q ) = b q D_{\mathbb{R}}u(q)=b_q D R u ( q ) = b q q q q (123) show that D R u D_{\mathbb{R}}u D R u M x M_x M x D R 2 u = M x D_{\mathbb{R}}^2u=M_x D R 2 u = M x (118) , using (120) , r 0 < 1 r_0<1 r 0 < 1 (114) , proves (115) .
A bounded trace and a Dini density give regularity
We prove the regularity statement used both in the compactness argument and when passing from continuous approximations to a weak solution.
Lemma 6.3. Let u u u Ω \Omega Ω
( − 1 ∂ ∂ ˉ u ) n = f ω 0 n (\sqrt{-1}\partial\bar\partial u)^n=f\omega_0^n ( − 1 ∂ ∂ ˉ u ) n = f ω 0 n in the Bedford–Taylor sense. Suppose f f f Δ R u \Delta_{\mathbb{R}}u Δ R u u ∈ C 2 ( Ω ) u\in C^2(\Omega) u ∈ C 2 ( Ω )
Proof. Work on a relatively compact ball on which f ≥ c 0 > 0 f\ge c_0>0 f ≥ c 0 > 0 tr D C 2 u ≤ H \operatorname{tr}D_{\mathbb{C}}^2u\le H tr D C 2 u ≤ H H > 0 H>0 H > 0 u ∈ W l o c 2 , p u\in W_{\mathrm{loc}}^{2,p} u ∈ W loc 2 , p p > 1 p>1 p > 1 0 ≤ D C 2 u ≤ H I 0\le D_{\mathbb{C}}^2u\le HI 0 ≤ D C 2 u ≤ H I
To identify the determinant, mollify on a smaller ball. Since u ∈ W l o c 2 , p u\in W_{\mathrm{loc}}^{2,p} u ∈ W loc 2 , p p > n p>n p > n L p L^p L p L p / n L^{p/n} L p / n
det D C 2 u = f almost everywhere . (124) \det D_{\mathbb{C}}^2u=f\quad\text{almost everywhere}. \tag*{(124)} det D C 2 u = f almost everywhere . (124) Combining (124) with f ≥ c 0 f\ge c_0 f ≥ c 0 0 ≤ D C 2 u ≤ H I 0\le D_{\mathbb{C}}^2u\le HI 0 ≤ D C 2 u ≤ H I
κ I ≤ D C 2 u ≤ H I , κ = c 0 H n − 1 . (125) \kappa I\le D_{\mathbb{C}}^2u\le HI,\qquad\kappa=\frac{c_0}{H^{n-1}}. \tag*{(125)} κ I ≤ D C 2 u ≤ H I , κ = H n − 1 c 0 . (125) We construct a smooth uniformly elliptic operator that agrees with log det \log\det log det (125) . The resulting equation will satisfy the hypotheses of the interior Dini estimate.
The nonempty Hessian range in (125) implies 0 < κ ≤ H 0<\kappa\le H 0 < κ ≤ H χ ∈ C ∞ ( ( κ / 4 , 4 H ) ) \chi\in C^\infty((\kappa/4,4H)) χ ∈ C ∞ (( κ /4 , 4 H )) 0 ≤ χ ≤ 1 0\le\chi\le1 0 ≤ χ ≤ 1 χ = 1 \chi=1 χ = 1 [ κ / 2 , 2 H ] [\kappa/2,2H] [ κ /2 , 2 H ] b ( s ) = χ ( s ) / s 2 b(s)=\chi(s)/s^2 b ( s ) = χ ( s ) / s 2 s ≤ 0 s\le0 s ≤ 0 ϕ \phi ϕ R \mathbb{R} R
ϕ ′ ( t ) = 1 H − ∫ H t b ( s ) d s , ϕ ( H ) = log H . \phi'(t)=\frac{1}{H}-\int_H^t b(s)\,ds,\qquad\phi(H)=\log H. ϕ ′ ( t ) = H 1 − ∫ H t b ( s ) d s , ϕ ( H ) = log H . For t ≥ H t\ge H t ≥ H ∫ H 4 H s − 2 d s = 3 / ( 4 H ) \int_H^{4H}s^{-2}\,ds=3/(4H) ∫ H 4 H s − 2 d s = 3/ ( 4 H ) t ≤ H t\le H t ≤ H ∫ κ / 4 H s − 2 d s = 4 / κ − 1 / H \int_{\kappa/4}^H s^{-2}\,ds=4/\kappa-1/H ∫ κ /4 H s − 2 d s = 4/ κ − 1/ H
1 4 H ≤ ϕ ′ ( t ) ≤ 4 κ , ϕ ′ ′ ( t ) = − b ( t ) ≤ 0 , ∣ ϕ ′ ′ ( t ) ∣ ≤ 16 κ 2 ( t ∈ R ) . (126) \frac{1}{4H}\le\phi'(t)\le\frac{4}{\kappa},\qquad\phi''(t)=-b(t)\le0,\qquad|\phi''(t)|\le\frac{16}{\kappa^2}\quad(t\in\mathbb{R}). \tag*{(126)} 4 H 1 ≤ ϕ ′ ( t ) ≤ κ 4 , ϕ ′′ ( t ) = − b ( t ) ≤ 0 , ∣ ϕ ′′ ( t ) ∣ ≤ κ 2 16 ( t ∈ R ) . (126) On [ κ / 2 , 2 H ] [\kappa/2,2H] [ κ /2 , 2 H ] H H H t t t χ = 1 \chi=1 χ = 1 ϕ ′ ( t ) = 1 / t \phi'(t)=1/t ϕ ′ ( t ) = 1/ t H H H ϕ ( t ) = log t \phi(t)=\log t ϕ ( t ) = log t
Use the projection H \mathcal{H} H
F ( N ) = tr ϕ ( H ( N ) ) , N ∈ Sym ( 2 n ) . (127) F(N)=\operatorname{tr}\phi(\mathcal{H}(N)),\qquad N\in\operatorname{Sym}(2n). \tag*{(127)} F ( N ) = tr ϕ ( H ( N )) , N ∈ Sym ( 2 n ) . (127) The function ϕ \phi ϕ H ( N ) \mathcal{H}(N) H ( N ) F F F A A A λ i \lambda_i λ i Q Q Q
D 2 ( tr ϕ ) ( A ) [ Q , Q ] = ∑ i ϕ ′ ′ ( λ i ) ∣ Q i i ˉ ∣ 2 + 2 ∑ i < j ϕ ′ ( λ i ) − ϕ ′ ( λ j ) λ i − λ j ∣ Q i j ˉ ∣ 2 . D^2(\operatorname{tr}\phi)(A)[Q,Q]=\sum_i\phi''(\lambda_i)|Q_{i\bar{i}}|^2+2\sum_{i<j}\frac{\phi'(\lambda_i)-\phi'(\lambda_j)}{\lambda_i-\lambda_j}|Q_{i\bar{j}}|^2. D 2 ( tr ϕ ) ( A ) [ Q , Q ] = i ∑ ϕ ′′ ( λ i ) ∣ Q i i ˉ ∣ 2 + 2 i < j ∑ λ i − λ j ϕ ′ ( λ i ) − ϕ ′ ( λ j ) ∣ Q i j ˉ ∣ 2 . At repeated eigenvalues, the quotient is interpreted as ϕ ′ ′ ( λ i ) \phi''(\lambda_i) ϕ ′′ ( λ i ) 32 / κ 2 32/\kappa^2 32/ κ 2 tr ϕ \operatorname{tr}\phi tr ϕ F F F
For a real positive semidefinite matrix P P P
D F ( N ) [ P ] = tr [ ϕ ′ ( H ( N ) ) H ( P ) ] . DF(N)[P]=\operatorname{tr}\left[\phi'(\mathcal{H}(N))\mathcal{H}(P)\right]. D F ( N ) [ P ] = tr [ ϕ ′ ( H ( N )) H ( P ) ] . The derivative bounds (6.21) and the positivity and trace identities in (6.1) imply
1 16 H tr R P ≤ D F ( N ) [ P ] ≤ 1 κ tr R P . (128) \frac{1}{16\mathcal{H}}\operatorname{tr}_{\mathbb{R}}P\le DF(N)[P]\le\frac{1}{\kappa}\operatorname{tr}_{\mathbb{R}}P. \tag*{(128)} 16 H 1 tr R P ≤ D F ( N ) [ P ] ≤ κ 1 tr R P . (128) Integrating along N + s P N+sP N + s P F F F
By (6.20), the complex Hessian of u u u ϕ = log \phi=\log ϕ = log
F ( D R 2 u ) = log det D C 2 u = log f (129) F(D^2_{\mathbb{R}}u)=\log\det D^2_{\mathbb{C}}u=\log f \tag*{(129)} F ( D R 2 u ) = log det D C 2 u = log f (129) almost everywhere. Since u ∈ W l o c 2 , p u\in W^{2,p}_{\mathrm{loc}} u ∈ W loc 2 , p p > 2 n p>2n p > 2 n [10 ] . On the ball under consideration, f ≥ c 0 f\ge c_0 f ≥ c 0
∣ log f ( x ) − log f ( y ) ∣ ≤ c 0 − 1 ∣ f ( x ) − f ( y ) ∣ . |\log f(x)-\log f(y)|\le c_0^{-1}|f(x)-f(y)|. ∣ log f ( x ) − log f ( y ) ∣ ≤ c 0 − 1 ∣ f ( x ) − f ( y ) ∣. Since f f f log f \log f log f M M M N ↦ F ( N + M ) − F ( M ) N\mapsto F(N+M)-F(M) N ↦ F ( N + M ) − F ( M ) [26 K82] therefore verifies the uniform homogeneous estimate required in [57 Section 3.1] . The interior Dini estimate, including its viscosity version, then gives u ∈ C 2 u\in C^2 u ∈ C 2 [57 Theorem 3.1 and Section 3.3(ii)] . The ellipticity and derivative bounds above depend only on n n n κ \kappa κ H \mathcal{H} H
A small averaging remainder gives a real Hessian bound
We prove the existence of a smallness threshold by letting the averaging remainder tend to zero in a compactness argument. Lemma 6.3 regularizes the limit, and Lemma 6.2 transfers its quadratic approximations to the approximating solutions.
Theorem 6.4 (From a pseudo-estimate to a C 2 C^2 C 2 Fix n n n K 0 K_0 K 0 A > 0 A>0 A > 0 0 < c 0 ≤ c 1 0<c_0\le c_1 0 < c 0 ≤ c 1 0 < t ∗ ≤ 1 / 4 0<t_*\le1/4 0 < t ∗ ≤ 1/4 ω \omega ω δ 0 > 0 \delta_0>0 δ 0 > 0 C < ∞ C<\infty C < ∞ u ∈ C ( B 2 ) u\in C(B_2) u ∈ C ( B 2 )
( − 1 ∂ ∂ ‾ u ) n = f ω 0 n , ∥ u ∥ L ∞ ( B 2 ) ≤ K 0 , c 0 ≤ f ≤ c 1 , (\sqrt{-1}\partial\overline{\partial}u)^n=f\omega_0^n,\qquad\lVert u\rVert_{L^\infty(B_2)}\le K_0,\qquad c_0\le f\le c_1, ( − 1 ∂ ∂ u ) n = f ω 0 n , ∥ u ∥ L ∞ ( B 2 ) ≤ K 0 , c 0 ≤ f ≤ c 1 , with ∣ f ( x ) − f ( y ) ∣ ≤ ω ( ∣ x − y ∣ ) |f(x)-f(y)|\le\omega(|x-y|) ∣ f ( x ) − f ( y ) ∣ ≤ ω ( ∣ x − y ∣ ) ∣ x − y ∣ ≤ 1 |x-y|\le1 ∣ x − y ∣ ≤ 1
0 ≤ G t u ( z ) − u ( z ) ≤ A t 2 + δ ( z ∈ B 3 / 2 , 0 < t < t ∗ ) , 0 ≤ δ ≤ δ 0 , (130) 0\le G_tu(z)-u(z)\le At^2+\delta\qquad(z\in B_{3/2},\ 0<t<t_*),\qquad0\le\delta\le\delta_0, \tag*{(130)} 0 ≤ G t u ( z ) − u ( z ) ≤ A t 2 + δ ( z ∈ B 3/2 , 0 < t < t ∗ ) , 0 ≤ δ ≤ δ 0 , (130) then u ∈ C 2 ( B 1 ) u\in C^2(B_1) u ∈ C 2 ( B 1 ) ∥ u ∥ C 2 ( B 1 ) ≤ C \lVert u\rVert_{C^2(B_1)}\le C ∥ u ∥ C 2 ( B 1 ) ≤ C
Proof. If no such threshold and bound exist, choose solutions u j u_j u j f j f_j f j δ j → 0 \delta_j\to0 δ j → 0 u j ∉ C 2 ( B 1 ) u_j\notin C^2(B_1) u j ∈ / C 2 ( B 1 ) ∥ u j ∥ C 2 ( B 1 ) > j \lVert u_j\rVert_{C^2(B_1)}>j ∥ u j ∥ C 2 ( B 1 ) > j t t t L 1 L^1 L 1 G t G_t G t u j u_j u j (130) shows that u j u_j u j A t 2 + δ j At^2+\delta_j A t 2 + δ j t t t j j j B 3 / 2 B_{3/2} B 3/2 f j f_j f j
u j ⟶ u ∞ , f j ⟶ f ∞ locally uniformly on B 3 / 2 . (131) u_j\longrightarrow u_\infty,\qquad f_j\longrightarrow f_\infty\quad\text{locally uniformly on }B_{3/2}. \tag*{(131)} u j ⟶ u ∞ , f j ⟶ f ∞ locally uniformly on B 3/2 . (131) The local uniform convergence (131) and Bedford–Taylor continuity give ( − 1 ∂ ∂ ‾ u ∞ ) n = f ∞ ω 0 n (\sqrt{-1}\partial\overline{\partial}u_\infty)^n=f_\infty\omega_0^n ( − 1 ∂ ∂ u ∞ ) n = f ∞ ω 0 n (130) , (131) , and δ j → 0 \delta_j\to0 δ j → 0 0 ≤ G t u ∞ − u ∞ ≤ A t 2 0\le G_tu_\infty-u_\infty\le At^2 0 ≤ G t u ∞ − u ∞ ≤ A t 2 t t t φ \varphi φ
∫ u ∞ G t φ − φ t 2 d ξ = ∫ φ G t u ∞ − u ∞ t 2 d ξ ≤ A ∫ φ d ξ . \int u_\infty\frac{G_t\varphi-\varphi}{t^2}\,d\xi=\int\varphi\frac{G_tu_\infty-u_\infty}{t^2}\,d\xi\le A\int\varphi\,d\xi. ∫ u ∞ t 2 G t φ − φ d ξ = ∫ φ t 2 G t u ∞ − u ∞ d ξ ≤ A ∫ φ d ξ . The expansion (2.2) therefore gives
0 ≤ Δ R u ∞ ≤ 8 ( n + 1 ) A in distributions on B 3 / 2 . (132) 0\le\Delta_{\mathbb{R}}u_\infty\le8(n+1)A\quad\text{in distributions on }B_{3/2}. \tag*{(132)} 0 ≤ Δ R u ∞ ≤ 8 ( n + 1 ) A in distributions on B 3/2 . (132) The limit equation, (132) , and Lemma 6.3 give u ∞ ∈ C 2 ( B 3 / 2 ) u_\infty\in C^2(B_{3/2}) u ∞ ∈ C 2 ( B 3/2 ) (125) , using H = 1 + 2 ( n + 1 ) A H=1+2(n+1)A H = 1 + 2 ( n + 1 ) A
Let E = B ‾ 5 / 4 E=\overline{B}_{5/4} E = B 5/4 P x P_x P x u ∞ u_\infty u ∞ x ∈ E x\in E x ∈ E
sup x ∈ E r − 2 ∥ u ∞ − P x ∥ L ∞ ( B r ( x ) ) ⟶ 0 ( r ↓ 0 ) . (133) \sup_{x\in E}r^{-2}\lVert u_\infty-P_x\rVert_{L^\infty(B_r(x))}\longrightarrow0\qquad(r\downarrow0). \tag*{(133)} x ∈ E sup r − 2 ∥ u ∞ − P x ∥ L ∞ ( B r ( x )) ⟶ 0 ( r ↓ 0 ) . (133) By (125) and (132) , choose 0 < λ ≤ Λ 0<\lambda\le\Lambda 0 < λ ≤ Λ
2 λ I ≤ D C 2 P x ≤ Λ 2 I ( x ∈ E ) . (134) 2\lambda I\le D^2_{\mathbb{C}}P_x\le\frac{\Lambda}{2}I\qquad(x\in E). \tag*{(134)} 2 λ I ≤ D C 2 P x ≤ 2 Λ I ( x ∈ E ) . (134) These margins leave room for the small corrections introduced below. We take the smallness threshold from Lemma 6.2 with parameters λ \lambda λ Λ \Lambda Λ r 0 < 1 / 8 r_0<1/8 r 0 < 1/8 (133) and D ω ( r 0 ) D_\omega(r_0) D ω ( r 0 ) f j ( x ) f_j(x) f j ( x ) det D C 2 P x = f ∞ ( x ) \det D^2_{\mathbb{C}}P_x=f_\infty(x) det D C 2 P x = f ∞ ( x ) (134) and (131) , this yields numbers a j , x → 0 a_{j,x}\to0 a j , x → 0 x ∈ E x\in E x ∈ E
P j , x = P x + a j , x ∣ z − x ∣ 2 , det D C 2 P j , x = f j ( x ) . (135) P_{j,x}=P_x+a_{j,x}|z-x|^2,\qquad\det D^2_{\mathbb{C}}P_{j,x}=f_j(x). \tag*{(135)} P j , x = P x + a j , x ∣ z − x ∣ 2 , det D C 2 P j , x = f j ( x ) . (135) Indeed, the determinant of D C 2 P x + a I D_{\mathbb{C}}^2P_x+aI D C 2 P x + a I a a a r 0 r_0 r 0
r 0 − 2 ∥ u j − P j , x ∥ L ∞ ( B r 0 ( x ) ) ≤ r 0 − 2 ∥ u j − u ∞ ∥ L ∞ ( B 11 / 8 ) + r 0 − 2 ∥ u ∞ − P x ∥ L ∞ ( B r 0 ( x ) ) + ∣ a j , x ∣ . (136) r_0^{-2}\|u_j-P_{j,x}\|_{L^\infty(B_{r_0}(x))}\leq r_0^{-2}\|u_j-u_\infty\|_{L^\infty(B_{11/8})}
+r_0^{-2}\|u_\infty-P_x\|_{L^\infty(B_{r_0}(x))}+|a_{j,x}|. \tag*{(136)} r 0 − 2 ∥ u j − P j , x ∥ L ∞ ( B r 0 ( x )) ≤ r 0 − 2 ∥ u j − u ∞ ∥ L ∞ ( B 11/8 ) + r 0 − 2 ∥ u ∞ − P x ∥ L ∞ ( B r 0 ( x )) + ∣ a j , x ∣. (136) By (6.26), the choice of r 0 r_0 r 0 a j , x → 0 a_{j,x}\to0 a j , x → 0 x ∈ E x\in E x ∈ E j j j u ∞ ∈ C 2 u_\infty\in C^2 u ∞ ∈ C 2 E E E P j , x P_{j,x} P j , x
λ I ≤ D C 2 P j , x ≤ Λ I . \lambda I\leq D_{\mathbb{C}}^2P_{j,x}\leq\Lambda I. λ I ≤ D C 2 P j , x ≤ Λ I . The radius r 0 r_0 r 0 u ∞ u_\infty u ∞ C 2 C^2 C 2 j j j B 5 / 4 B_{5/4} B 5/4 ∥ u j ∥ ∞ ≤ K 0 \|u_j\|_\infty\leq K_0 ∥ u j ∥ ∞ ≤ K 0 B 1 B_1 B 1
Apply the criterion to the distant-boundary remainder
Proof of Theorem 1.3. We apply Theorem 6.4 to the continuous exact-density replacements from Section 5. Choose the parameters of the positive-density estimate once from the fixed growth buffers and density bounds used there. On B 2 B_2 B 2 W , K 0 < ∞ W,K_0<\infty W , K 0 < ∞
0 < V j = v j + J − a ρ A h ≤ W , ∥ v j ∥ L ∞ ( B 2 ) ≤ K 0 0<V_j=v_j+J-a\rho Ah\leq W,\qquad\|v_j\|_{L^\infty(B_2)}\leq K_0 0 < V j = v j + J − a ρ A h ≤ W , ∥ v j ∥ L ∞ ( B 2 ) ≤ K 0 for every sufficiently large j j j j j j L L L A ∗ = 2 W ε + A_* = 2W\varepsilon_+ A ∗ = 2 W ε + t ∗ = 1 / 4 t_* = 1/4 t ∗ = 1/4 n n n K 0 K_0 K 0 A ∗ A_* A ∗ c 0 c_0 c 0 c 1 c_1 c 1 ω \omega ω t ∗ t_* t ∗ δ 0 > 0 \delta_0>0 δ 0 > 0 L 0 L_0 L 0 B 2 B_2 B 2
E + W q L 0 ≤ δ 0 . (137) E_+Wq_{L_0}\leq\delta_0. \tag*{(137)} E + W q L 0 ≤ δ 0 . (137) The formula (3.13) gives q L ↓ 0 q_L\downarrow0 q L ↓ 0 L ≥ L 0 L\geq L_0 L ≥ L 0 v j ↓ u v_j\downarrow u v j ↓ u B L + 1 B_{L+1} B L + 1
0 ≤ G t v j − v j ≤ A ∗ t 2 + E + W q L ≤ A ∗ t 2 + δ 0 on B 3 / 2 , 0 < t < t ∗ . 0\leq G_tv_j-v_j\leq A_*t^2+E_+Wq_L\leq A_*t^2+\delta_0\quad\text{on }B_{3/2},\qquad0<t<t_*. 0 ≤ G t v j − v j ≤ A ∗ t 2 + E + W q L ≤ A ∗ t 2 + δ 0 on B 3/2 , 0 < t < t ∗ . Together with the fixed density bounds and modulus, this verifies Theorem 6.4, which yields ∥ v j ∥ C 2 ( B 1 ) ≤ H \|v_j\|_{C^2(B_1)}\leq H ∥ v j ∥ C 2 ( B 1 ) ≤ H H H H j , L j,L j , L
These bounds and the pointwise convergence imply local C 1 C^1 C 1 u ∈ C l o c 1 , 1 ( B 1 ) u\in C^{1,1}_{\mathrm{loc}}(B_1) u ∈ C loc 1 , 1 ( B 1 ) u ∈ C 2 ( B 1 ) u\in C^2(B_1) u ∈ C 2 ( B 1 )
Proof of Corollary 1.4. Theorem 1.3 gives u ∈ C 2 ( B 1 ) u\in C^2(B_1) u ∈ C 2 ( B 1 ) A = D C 2 u A=D_{\mathbb{C}}^2u A = D C 2 u u ∈ C 2 u\in C^2 u ∈ C 2 f f f det A = f \det A=f det A = f A ≥ 0 A\geq0 A ≥ 0 det A = f ≥ c 0 > 0 \det A=f\geq c_0>0 det A = f ≥ c 0 > 0 A A A C 2 C^2 C 2 f ≥ c 0 f \ge c_0 f ≥ c 0 f f f [58 Theorem 1.1] , applied on smaller balls, gives u ∈ C l o c 2 , β ( B 1 ) u \in C^{2,\beta}_{\mathrm{loc}}(B_1) u ∈ C loc 2 , β ( B 1 ) β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 )
To obtain one more derivative, fix a real coordinate direction e e e v h ( z ) = ( u ( z + h e ) − u ( z ) ) / h v_h(z)=(u(z+he)-u(z))/h v h ( z ) = ( u ( z + h e ) − u ( z )) / h h ≠ 0 h\ne0 h = 0 log det \log\det log det A ( z ) A(z) A ( z ) A ( z + h e ) A(z+he) A ( z + h e )
tr ( a h ( z ) D C 2 v h ( z ) ) = log f ( z + h e ) − log f ( z ) h , \operatorname{tr}\left(a_h(z)D_{\mathbb{C}}^2v_h(z)\right)=\frac{\log f(z+he)-\log f(z)}{h}, tr ( a h ( z ) D C 2 v h ( z ) ) = h log f ( z + h e ) − log f ( z ) , a h ( z ) = ∫ 0 1 ( ( 1 − t ) A ( z ) + t A ( z + h e ) ) − 1 d t . (138) a_h(z)=\int_0^1\left((1-t)A(z)+tA(z+he)\right)^{-1}\,dt. \tag*{(138)} a h ( z ) = ∫ 0 1 ( ( 1 − t ) A ( z ) + t A ( z + h e ) ) − 1 d t . (138) Theorem 1.3 and det A = f ≥ c 0 \det A=f\ge c_0 det A = f ≥ c 0 A A A A ∈ C l o c β ( B 1 ) A\in C^\beta_{\mathrm{loc}}(B_1) A ∈ C loc β ( B 1 ) a h a_h a h C β C^\beta C β h h h f f f C β C^\beta C β (138) , and u ∈ C 1 u\in C^1 u ∈ C 1 v h v_h v h [29 Chapter 6] therefore bound v h v_h v h C 2 , β C^{2,\beta} C 2 , β h → 0 h\to0 h → 0 ∂ e u ∈ C l o c 2 , β ( B 1 ) \partial_eu\in C^{2,\beta}_{\mathrm{loc}}(B_1) ∂ e u ∈ C loc 2 , β ( B 1 ) A − 1 A^{-1} A − 1 C 1 , β C^{1,\beta} C 1 , β log f \log f log f [29 Chapter 6] gives derivatives of every order. Thus u ∈ C ∞ ( B 1 ) u\in C^\infty(B_1) u ∈ C ∞ ( B 1 )
Proof of the Liouville theorem
We first use the C 2 C^2 C 2
Smoothness and the trace bound
Let u u u Theorem 1.1 . For each a ∈ C n a\in\mathbb{C}^n a ∈ C n ρ ( z + a ) / ρ ( z ) \rho(z+a)/\rho(z) ρ ( z + a ) / ρ ( z ) ψ ( z + a ) / ψ ( z ) \psi(z+a)/\psi(z) ψ ( z + a ) / ψ ( z ) ∣ z ∣ → ∞ |z|\to\infty ∣ z ∣ → ∞ a a a (4) gives the same growth condition for u a ( z ) = u ( z + a ) u_a(z)=u(z+a) u a ( z ) = u ( z + a ) 0 < c a ≤ 1 0<c_a\le1 0 < c a ≤ 1 C a ≥ 1 C_a\ge1 C a ≥ 1 α \alpha α (3) , so Theorem 1.3 and Corollary 1.4, applied to u a u_a u a f ≡ 1 f\equiv1 f ≡ 1 u a ∈ C ∞ ( B 1 ) u_a\in C^\infty(B_1) u a ∈ C ∞ ( B 1 ) a a a u ∈ C ∞ ( C n ) u\in C^\infty(\mathbb{C}^n) u ∈ C ∞ ( C n )
Now apply Theorem 1.5 to u u u c , C c,C c , C K = 0 K=0 K = 0 q L → 0 q_L\to0 q L → 0 (32) , letting L → ∞ L\to\infty L → ∞ z , t z,t z , t (13) . Smoothness and the expansion (19) then give
tr D C 2 u ( z ) = 2 ( n + 1 ) lim t → 0 G t u ( z ) − u ( z ) t 2 ≤ 4 ( n + 1 ) V ( z ) ε − . (139) \operatorname{tr}D_{\mathbb{C}}^2u(z)=2(n+1)\lim_{t\to0}\frac{\mathcal{G}_tu(z)-u(z)}{t^2}\le4(n+1)V(z)\varepsilon^{-}. \tag*{(139)} tr D C 2 u ( z ) = 2 ( n + 1 ) t → 0 lim t 2 G t u ( z ) − u ( z ) ≤ 4 ( n + 1 ) V ( z ) ε − . (139) This proves (14) and Corollary 1.6 .
Make the complex Hessian constant
The normalization (36) , with s = log ρ A s=\log\rho_A s = log ρ A (31) , gives V ≤ C α ρ A ( 1 + log ρ A ) 1 − α V\le C_{\alpha}\rho_A(1+\log\rho_A)^{1-\alpha} V ≤ C α ρ A ( 1 + log ρ A ) 1 − α (14) implies that for every θ > 0 \theta>0 θ > 0 H θ < ∞ H_\theta<\infty H θ < ∞
tr D C 2 u ≤ H θ ( 1 + ∣ z ∣ 2 ) ε − + θ . (140) \operatorname{tr}D_{\mathbb{C}}^2u\le H_\theta(1+|z|^2)^{\varepsilon_{-}+\theta}. \tag*{(140)} tr D C 2 u ≤ H θ ( 1 + ∣ z ∣ 2 ) ε − + θ . (140) Here we used ( log r ) b ≤ C b , θ r θ (\log r)^b \le C_{b,\theta}r^\theta ( log r ) b ≤ C b , θ r θ r ≥ e r \ge e r ≥ e b = ε − ( 1 − α ) b=\varepsilon_-(1-\alpha) b = ε − ( 1 − α )
The quantitative Calabi estimate [37 Proposition 2.4] , applied to u R ( z ) = R − 2 u ( R z ) u_R(z)=R^{-2}u(Rz) u R ( z ) = R − 2 u ( R z ) B 2 B_2 B 2
sup B R ∣ D R ( u i ˉ j ) ∣ ≤ C n R ( sup B 2 R tr D C 2 u ) n + 1 ≤ C θ R − 1 + 2 ( n + 1 ) ( ε − + θ ) . (141) \sup_{B_R} \lvert D_R(u_{\bar{i}j})\rvert\le\frac{C_n}{R}\left(\sup_{B_{2R}}\operatorname{tr}D_{\mathbb{C}}^2u\right)^{n+1}\le C_\theta R^{-1+2(n+1)(\varepsilon_-+\theta)}. \tag*{(141)} B R sup ∣ D R ( u i ˉ j )∣ ≤ R C n ( B 2 R sup tr D C 2 u ) n + 1 ≤ C θ R − 1 + 2 ( n + 1 ) ( ε − + θ ) . (141) With ε − = 1 / [ 4 ( n + 1 ) ] \varepsilon_-=1/[4(n+1)] ε − = 1/ [ 4 ( n + 1 )] 0 < θ < 1 / [ 4 ( n + 1 ) ] 0<\theta<1/[4(n+1)] 0 < θ < 1/ [ 4 ( n + 1 )] R → ∞ R\to\infty R → ∞ (141) shows that D C 2 u = A D_{\mathbb{C}}^2u=A D C 2 u = A
Recover the quadratic potential
By (141) , D C 2 u = A D_{\mathbb{C}}^2u=A D C 2 u = A v = u − A [ z ] v=u-A[z] v = u − A [ z ] D C 2 v = 0 D_{\mathbb{C}}^2v=0 D C 2 v = 0 v v v v ≥ − A [ z ] ≥ − C ∗ ( 1 + ∣ z ∣ 2 ) v\ge-A[z]\ge-C_*(1+\lvert z\rvert^2) v ≥ − A [ z ] ≥ − C ∗ ( 1 + ∣ z ∣ 2 ) x x x
∫ B R ( x ) ∣ v ∣ = ∣ B R ∣ v ( x ) + 2 ∫ B R ( x ) v − ≤ C x R 2 n + 2 ( R ≥ 1 ) . (142) \int_{B_R(x)}\lvert v\rvert=\lvert B_R\rvert v(x)+2\int_{B_R(x)}v^-\le C_xR^{2n+2}\qquad(R\ge1). \tag*{(142)} ∫ B R ( x ) ∣ v ∣ = ∣ B R ∣ v ( x ) + 2 ∫ B R ( x ) v − ≤ C x R 2 n + 2 ( R ≥ 1 ) . (142) The interior harmonic derivative estimate applied to (142) yields
∣ D R 3 v ( x ) ∣ ≤ C n R − 2 n − 3 ∫ B R ( x ) ∣ v ∣ ≤ C x / R ⟶ 0. \lvert D_{\mathbb{R}}^3v(x)\rvert\le C_nR^{-2n-3}\int_{B_R(x)}\lvert v\rvert\le C_x/R\longrightarrow0. ∣ D R 3 v ( x )∣ ≤ C n R − 2 n − 3 ∫ B R ( x ) ∣ v ∣ ≤ C x / R ⟶ 0. Thus v v v u u u
An alternative geometric proof for the quadratic-growth special case
This appendix gives an alternative proof of the Liouville theorem for smooth entire solutions with two-sided quadratic growth. We obtain a global complex-Hessian bound from a Green-function probability measure supported where the metric is controlled. A Calabi estimate then makes the metric constant, and a harmonic estimate recovers the potential. The Bedford–Taylor case still requires the regularity established in Sections 5–7; the argument below can also be used once that regularity is known.
Theorem A.1 (Smooth quadratic-growth rigidity). Let u ∈ C ∞ ( C n ) u\in C^\infty(\mathbb{C}^n) u ∈ C ∞ ( C n )
det D C 2 u = 1 , m ρ ≤ u ≤ M ρ , ρ = 1 + ∣ z ∣ 2 , 0 < m ≤ M . \det D_{\mathbb{C}}^2u=1,\qquad m\rho\le u\le M\rho,\qquad\rho=1+\lvert z\rvert^2,\qquad0<m\le M. det D C 2 u = 1 , m ρ ≤ u ≤ M ρ , ρ = 1 + ∣ z ∣ 2 , 0 < m ≤ M . Then D C 2 u D_{\mathbb{C}}^2u D C 2 u u u u
Write g = D C 2 u g=D_{\mathbb{C}}^2u g = D C 2 u g 0 = I n g_0=I_n g 0 = I n D D D
Δ g φ = tr ( g − 1 D C 2 φ ) , S = tr ( g − 1 g 0 ) . (143) \Delta_g\varphi=\operatorname{tr}(g^{-1}D_{\mathbb{C}}^2\varphi),\qquad S=\operatorname{tr}(g^{-1}g_0). \tag*{(143)} Δ g φ = tr ( g − 1 D C 2 φ ) , S = tr ( g − 1 g 0 ) . (143) Thus Δ g \Delta_g Δ g Δ R \Delta_{\mathbb{R}} Δ R C C C n , m n,m n , m
Construct a probability measure on a controlled region
We construct a probability measure μ \mu μ S S S φ \varphi φ ∣ φ ∣ ≤ C ϱ |\varphi| \le C\varrho ∣ φ ∣ ≤ C ϱ Δ g φ ≥ 0 \Delta_g\varphi\ge0 Δ g φ ≥ 0 φ ( x ) \varphi(x) φ ( x ) u u u
By the determinant condition in Theorem A.1 , the arithmetic–geometric mean inequality gives S ≥ n S \ge n S ≥ n
Δ g u = n , Δ g ϱ = S , ∣ ∂ ϱ ∣ g 2 ≤ S ϱ . (144) \Delta_g u = n,\qquad\Delta_g\varrho= S,\qquad|\partial\varrho|_g^2 \le S\varrho. \tag*{(144)} Δ g u = n , Δ g ϱ = S , ∣ ∂ ϱ ∣ g 2 ≤ S ϱ . (144) To construct a positive barrier, set ℓ = max { n , 4 n / m } \ell=\max\{n,4n/m\} ℓ = max { n , 4 n / m } χ : [ n , ∞ ) → [ 0 , 1 ] \chi:[n,\infty)\to[0,1] χ : [ n , ∞ ) → [ 0 , 1 ] [ n , ℓ ] [n,\ell] [ n , ℓ ] [ 2 ℓ , ∞ ) [2\ell,\infty) [ 2 ℓ , ∞ )
V = u − m 2 ∣ z ∣ 2 , k = 2 χ ( S ) m ϱ . (145) V=u-\frac{m}{2}|z|^2,\qquad k=\frac{2\chi(S)}{m\varrho}. \tag*{(145)} V = u − 2 m ∣ z ∣ 2 , k = m ϱ 2 χ ( S ) . (145) The lower growth bound in Theorem A.1 gives V ≥ m ϱ / 2 V\ge m\varrho/2 V ≥ m ϱ /2 S ≤ ℓ S\le\ell S ≤ ℓ χ ( S ) = 1 \chi(S)=1 χ ( S ) = 1 k V ≥ 1 kV\ge1 k V ≥ 1 n / S ≤ 1 n/S\le1 n / S ≤ 1 S ≥ ℓ S\ge\ell S ≥ ℓ n / S ≤ m / 4 n/S\le m/4 n / S ≤ m /4 k V ≥ 0 kV\ge0 k V ≥ 0 Δ g V = n − m S / 2 \Delta_gV=n-mS/2 Δ g V = n − m S /2
( − Δ g + k S ) V ≥ m 4 S . (146) (-\Delta_g+kS)V\ge\frac{m}{4}S. \tag*{(146)} ( − Δ g + k S ) V ≥ 4 m S . (146) Fix x ∈ C n x\in\mathbb{C}^n x ∈ C n L L L x ∈ Ω L = { ϱ < L 2 } x\in\Omega_L=\{\varrho<L^2\} x ∈ Ω L = { ϱ < L 2 } G L ( x , y ) G_L(x,y) G L ( x , y ) − Δ g + k S -\Delta_g+kS − Δ g + k S ( − Δ g , y + k S ( y ) ) G L ( x , y ) = δ x (-\Delta_{g,y}+kS(y))G_L(x,y)=\delta_x ( − Δ g , y + k S ( y )) G L ( x , y ) = δ x d vol g ( y ) d\operatorname{vol}_g(y) d vol g ( y )
d ν L ( y ) = S ( y ) G L ( x , y ) d vol g ( y ) , d\nu_L(y)=S(y)G_L(x,y)\,d\operatorname{vol}_g(y), d ν L ( y ) = S ( y ) G L ( x , y ) d vol g ( y ) , and let d β L d\beta_L d β L k S ≥ 0 kS\ge0 k S ≥ 0
φ ( x ) = ∫ Ω L ( k φ − Δ g φ S ) d ν L + ∫ ∂ Ω L φ d β L (147) \varphi(x)=\int_{\Omega_L}\left(k\varphi-\frac{\Delta_g\varphi}{S}\right)d\nu_L+\int_{\partial\Omega_L}\varphi\,d\beta_L \tag*{(147)} φ ( x ) = ∫ Ω L ( k φ − S Δ g φ ) d ν L + ∫ ∂ Ω L φ d β L (147) for smooth φ \varphi φ Ω ‾ L \overline{\Omega}_L Ω L V V V
ν L ( C n ) ≤ 4 V ( x ) m . (148) \nu_L(\mathbb{C}^n)\le\frac{4V(x)}{m}. \tag*{(148)} ν L ( C n ) ≤ m 4 V ( x ) . (148) Here the measures on Ω L \Omega_L Ω L ν L \nu_L ν L ν \nu ν
Testing (A.5) with 1 1 1 ϱ \varrho ϱ Δ g ϱ = S \Delta_g\varrho=S Δ g ϱ = S ϱ = L 2 \varrho=L^2 ϱ = L 2 ∂ Ω L \partial\Omega_L ∂ Ω L
∫ k d ν L + ∫ d β L = 1 , (149) \int k\,d\nu_L+\int d\beta_L=1, \tag*{(149)} ∫ k d ν L + ∫ d β L = 1 , (149) ∫ k ϱ d ν L + L 2 ∫ d β L = ϱ ( x ) + ν L ( C n ) . (150) \int k\varrho\,d\nu_L+L^2\int d\beta_L=\varrho(x)+\nu_L(\mathbb{C}^n). \tag*{(150)} ∫ k ϱ d ν L + L 2 ∫ d β L = ϱ ( x ) + ν L ( C n ) . (150) To control the second boundary term, choose ϑ ∈ C ∞ ( [ 0 , ∞ ) ) \vartheta\in C^\infty([0,\infty)) ϑ ∈ C ∞ ([ 0 , ∞ )) 0 ≤ ϑ ≤ 1 0\le\vartheta\le1 0 ≤ ϑ ≤ 1 [ 0 , 1 / 4 ] [0,1/4] [ 0 , 1/4 ] [ 3 / 4 , ∞ ) [3/4,\infty) [ 3/4 , ∞ )
ϑ L = L 2 ϑ ( ϱ / L 2 ) . (151) \vartheta_L=L^2\vartheta(\varrho/L^2). \tag*{(151)} ϑ L = L 2 ϑ ( ϱ / L 2 ) . (151) For large L L L ϑ L ( x ) = 0 \vartheta_L(x)=0 ϑ L ( x ) = 0 ϑ L = L 2 \vartheta_L=L^2 ϑ L = L 2 ∂ Ω L \partial\Omega_L ∂ Ω L
Δ g ϑ L S = ϑ ′ ( ϱ / L 2 ) + ϑ ′ ′ ( ϱ / L 2 ) L 2 ∣ ∂ ϱ ∣ g 2 S ≤ C 1 { ϱ ≥ L 2 / 4 } on Ω L . (152) \frac{\Delta_g\vartheta_L}{S}=\vartheta'(\varrho/L^2)+\frac{\vartheta''(\varrho/L^2)}{L^2}\frac{|\partial\varrho|_g^2}{S}\le C\mathbf{1}_{\{\varrho\ge L^2/4\}}\quad\text{on }\Omega_L. \tag*{(152)} S Δ g ϑ L = ϑ ′ ( ϱ / L 2 ) + L 2 ϑ ′′ ( ϱ / L 2 ) S ∣ ∂ ϱ ∣ g 2 ≤ C 1 { ϱ ≥ L 2 /4 } on Ω L . (152) Insert ϑ L \vartheta_L ϑ L k ϑ L ≥ 0 k\vartheta_L\ge0 k ϑ L ≥ 0 ν L ≤ ν \nu_L\le\nu ν L ≤ ν
0 ≤ L 2 ∫ d β L ≤ C ν ( { ϱ ≥ L 2 / 4 } ) ⟶ 0. (153) 0\le L^2\int d\beta_L\le C\nu(\{\varrho\ge L^2/4\})\longrightarrow0. \tag*{(153)} 0 ≤ L 2 ∫ d β L ≤ C ν ({ ϱ ≥ L 2 /4 }) ⟶ 0. (153) The last limit follows from finiteness of ν \nu ν
Define d μ = k d ν d\mu=k\,d\nu d μ = k d ν ν = ν x \nu=\nu_x ν = ν x μ = μ x \mu=\mu_x μ = μ x x x x
μ ( C n ) = 1 , ∫ ϱ d μ = ϱ ( x ) + ν ( C n ) < ∞ . (154) \mu(\mathbb{C}^n)=1,\qquad\int\varrho\,d\mu=\varrho(x)+\nu(\mathbb{C}^n)<\infty. \tag*{(154)} μ ( C n ) = 1 , ∫ ϱ d μ = ϱ ( x ) + ν ( C n ) < ∞. (154) By (A.3), the cutoff defining k k k S ≥ 2 ℓ S\ge2\ell S ≥ 2 ℓ d μ = k d ν d\mu=k\,d\nu d μ = k d ν { S ≤ 2 ℓ } \{S\le2\ell\} { S ≤ 2 ℓ } S S S g g g ( 2 ℓ ) − 1 (2\ell)^{-1} ( 2 ℓ ) − 1
( 2 ℓ ) − 1 g 0 ≤ g ≤ ( 2 ℓ ) n − 1 g 0 on supp μ . (155) (2\ell)^{-1}g_0\le g\le(2\ell)^{n-1}g_0\quad\text{on }\operatorname{supp}\mu. \tag*{(155)} ( 2 ℓ ) − 1 g 0 ≤ g ≤ ( 2 ℓ ) n − 1 g 0 on supp μ . (155) We will use the limiting Green formula for a smooth real function φ \varphi φ ∣ φ ∣ ≤ C ϱ |\varphi|\le C\varrho ∣ φ ∣ ≤ C ϱ Δ g φ ≥ 0 \Delta_g\varphi\ge0 Δ g φ ≥ 0 μ \mu μ Δ g φ / S \Delta_g\varphi/S Δ g φ / S L L L
φ ( x ) = ∫ φ d μ − ∫ Δ g φ S d ν . (156) \varphi(x)=\int\varphi\,d\mu-\int\frac{\Delta_g\varphi}{S}\,d\nu. \tag*{(156)} φ ( x ) = ∫ φ d μ − ∫ S Δ g φ d ν . (156) Use translations and disc averages to bound the metric
We now use the probability measure to compare the metric at x x x
For a flat displacement h ∈ C n h\in\mathbb{C}^n h ∈ C n
J ( h ) = ∫ u ( y + h ) d μ ( y ) . J(h)=\int u(y+h)\,d\mu(y). J ( h ) = ∫ u ( y + h ) d μ ( y ) . On bounded sets of h h h 0 < u ( y + h ) ≤ C ϱ ( y ) 0<u(y+h)\le C\varrho(y) 0 < u ( y + h ) ≤ C ϱ ( y ) Δ g [ u ( ⋅ + h ) ] ≥ 0 \Delta_g[u(\cdot+h)]\ge0 Δ g [ u ( ⋅ + h )] ≥ 0
J ( h ) − u ( x + h ) = ∫ tr ( g ( y ) − 1 g ( y + h ) ) d ν ( y ) S ( y ) . (157) J(h)-u(x+h)=\int\operatorname{tr}\bigl(g(y)^{-1}g(y+h)\bigr)\frac{d\nu(y)}{S(y)}. \tag*{(157)} J ( h ) − u ( x + h ) = ∫ tr ( g ( y ) − 1 g ( y + h ) ) S ( y ) d ν ( y ) . (157) Both sides are finite; the preceding bound u ( y + h ) ≤ C ϱ ( y ) u(y+h)\le C\varrho(y) u ( y + h ) ≤ C ϱ ( y ) h h h tr ( g ( y ) − 1 g ( y + h ) ) ≥ n \operatorname{tr}(g(y)^{-1}g(y+h))\ge n tr ( g ( y ) − 1 g ( y + h )) ≥ n h = 0 h=0 h = 0 h = 0 h=0 h = 0 h h h
J ( h ) − u ( x + h ) ≥ J ( 0 ) − u ( x ) . (158) J(h)-u(x+h)\ge J(0)-u(x). \tag*{(158)} J ( h ) − u ( x + h ) ≥ J ( 0 ) − u ( x ) . (158) We next justify the required second-order averaging. On supp μ \operatorname{supp}\mu supp μ S ≤ 2 ℓ S \le2\ell S ≤ 2 ℓ
tr ( g 0 − 1 g ( y + h ) ) ≤ ( 2 ℓ ) n tr ( g ( y ) − 1 g ( y + h ) ) S ( y ) . \operatorname{tr}(g_0^{-1}g(y+h)) \le(2\ell)^n \frac{\operatorname{tr}(g(y)^{-1}g(y+h))}{S(y)}. tr ( g 0 − 1 g ( y + h )) ≤ ( 2 ℓ ) n S ( y ) tr ( g ( y ) − 1 g ( y + h )) . Using d μ = k d ν d\mu=k\,d\nu d μ = k d ν k ≤ 2 / ( m ϱ ) k \le2/(m\varrho) k ≤ 2/ ( m ϱ )
∫ { ϱ > L 2 } tr ( g 0 − 1 g ( y + h ) ) d μ ( y ) ≤ C L 2 . (159) \int_{\{\varrho>L^2\}} \operatorname{tr}(g_0^{-1}g(y+h))\,d\mu(y) \le\frac{C}{L^2}. \tag*{(159)} ∫ { ϱ > L 2 } tr ( g 0 − 1 g ( y + h )) d μ ( y ) ≤ L 2 C . (159) uniformly for bounded h h h C C C n , m , M n,m,M n , m , M x x x ∣ h ∣ |h| ∣ h ∣ L L L h h h ϱ ≥ 1 \varrho\ge1 ϱ ≥ 1 y y y h h h ∫ g ( y + h ) d μ ( y ) \int g(y+h)\,d\mu(y) ∫ g ( y + h ) d μ ( y )
Fix a constant complex vector v v v
1 2 π ∫ 0 2 π u ( y + r e i θ v ) d θ − u ( y ) = 2 π ∫ ∣ ζ ∣ < r log r ∣ ζ ∣ g ( y + ζ v ) [ v ] d A ( ζ ) . (160) \frac{1}{2\pi}\int_0^{2\pi}u(y+re^{i\theta}v)\,d\theta-u(y)=\frac{2}{\pi}\int_{|\zeta|<r}\log\frac{r}{|\zeta|}\,g(y+\zeta v)[v]\,dA(\zeta). \tag*{(160)} 2 π 1 ∫ 0 2 π u ( y + r e i θ v ) d θ − u ( y ) = π 2 ∫ ∣ ζ ∣ < r log ∣ ζ ∣ r g ( y + ζ v ) [ v ] d A ( ζ ) . (160) The scalar kernel on the right has total mass r 2 r^2 r 2 h = r e i θ v h=re^{i\theta}v h = r e i θ v y y y x x x
0 ≤ 2 π ∫ ∣ ζ ∣ < r log r ∣ ζ ∣ ( ∫ g ( y + ζ v ) [ v ] d μ ( y ) − g ( x + ζ v ) [ v ] ) d A ( ζ ) . (161) 0 \le\frac{2}{\pi}\int_{|\zeta|<r}\log\frac{r}{|\zeta|}\left(\int g(y+\zeta v)[v]\,d\mu(y)-g(x+\zeta v)[v]\right)\,dA(\zeta). \tag*{(161)} 0 ≤ π 2 ∫ ∣ ζ ∣ < r log ∣ ζ ∣ r ( ∫ g ( y + ζ v ) [ v ] d μ ( y ) − g ( x + ζ v ) [ v ] ) d A ( ζ ) . (161) Divide (A.19) by r 2 r^2 r 2 r ↓ 0 r \downarrow0 r ↓ 0 r 2 r^2 r 2
g ( x ) [ v ] ≤ ∫ g ( y ) [ v ] d μ ( y ) . (162) g(x)[v] \le\int g(y)[v]\,d\mu(y). \tag*{(162)} g ( x ) [ v ] ≤ ∫ g ( y ) [ v ] d μ ( y ) . (162) Since v v v
g ( x ) ≤ ∫ g d μ ≤ ( 2 ℓ ) n − 1 g 0 . (163) g(x) \le\int g\,d\mu\le(2\ell)^{n-1}g_0. \tag*{(163)} g ( x ) ≤ ∫ g d μ ≤ ( 2 ℓ ) n − 1 g 0 . (163) The point x x x Theorem A.1 gives a lower eigenvalue bound as well. Thus
C − 1 g 0 ≤ g ≤ C g 0 on C n . (164) C^{-1}g_0 \le g \le Cg_0 \quad\text{on }\mathbb{C}^n. \tag*{(164)} C − 1 g 0 ≤ g ≤ C g 0 on C n . (164) In particular, g g g μ \mu μ
Apply the Calabi estimate to make the metric constant
The aim here is to eliminate the derivatives of the metric. The Calabi identities control a nonnegative third-derivative quantity and the bounded trace of g g g a − tr ( g 0 − 1 g ) a-\operatorname{tr}(g_0^{-1}g) a − tr ( g 0 − 1 g ) a > 1 + sup C n tr ( g 0 − 1 g ) a>1+\sup_{\mathbb{C}^n}\operatorname{tr}(g_0^{-1}g) a > 1 + sup C n tr ( g 0 − 1 g )
Let ∇ \nabla ∇ g g g B = ∇ − D B=\nabla-D B = ∇ − D
B i j k = ∑ ℓ ( g − 1 ) ℓ k D i g j ℓ ˉ , E = ∣ B ∣ g 2 , T = tr ( g 0 − 1 g ) . (165) B^k_{ij}=\sum_\ell(g^{-1})_{\ell k}D_i g_{j\bar\ell},\qquad E=|B|_g^2,\qquad T=\operatorname{tr}(g_0^{-1}g). \tag*{(165)} B ij k = ℓ ∑ ( g − 1 ) ℓ k D i g j ℓ ˉ , E = ∣ B ∣ g 2 , T = tr ( g 0 − 1 g ) . (165) The determinant equation in Theorem A.1 gives Ric ( g ) = 0 \operatorname{Ric}(g)=0 Ric ( g ) = 0 T T T
Δ g E ≥ 0 , Δ g T ≥ c E , ∣ ∂ T ∣ g 2 ≤ C E , (166) \Delta_g E \ge0,\qquad\Delta_g T \ge cE,\qquad\lvert\partial T\rvert_g^2 \le CE, \tag*{(166)} Δ g E ≥ 0 , Δ g T ≥ c E , ∣ ∂ T ∣ g 2 ≤ C E , (166) where c > 0 c>0 c > 0 D D D ∇ ‾ b ˉ i j B i j k = − R i b ˉ j k \overline{\nabla}_{\bar b_{ij}}B^k_{ij}=-R^k_{i\bar b j} ∇ b ˉ ij B ij k = − R i b ˉ j k Theorem A.1 gives Ric ( g ) = 0 \operatorname{Ric}(g)=0 Ric ( g ) = 0 ∣ B ∣ g 2 \lvert B\rvert_g^2 ∣ B ∣ g 2 [53 Section 2.4] . Differentiating log det g = 0 \log\det g=0 log det g = 0 Δ g T \Delta_gT Δ g T D g Dg D g g 0 − 1 g_0^{-1} g 0 − 1 E E E g 0 − 1 g_0^{-1} g 0 − 1 D i T = tr ( g 0 − 1 D i g ) D_iT=\operatorname{tr}(g_0^{-1}D_ig) D i T = tr ( g 0 − 1 D i g )
Choose a > 1 + sup T a>1+\sup T a > 1 + sup T
q = E a − T . (167) q=\frac{E}{a-T}. \tag*{(167)} q = a − T E . (167) The denominator is bounded above and below by positive constants. Applying the product rule to E = ( a − T ) q E=(a-T)q E = ( a − T ) q
Δ g q ≥ c 1 q 2 + 2 q a − T Re ⟨ ∂ T , ∂ q ⟩ g , ∣ ∂ T ∣ g a − T ≤ C q (168) \Delta_gq\ge c_1q^2+\frac{2q}{a-T}\operatorname{Re}\langle\partial T,\partial q\rangle_g,\qquad\frac{\lvert\partial T\rvert_g}{a-T}\le C\sqrt q \tag*{(168)} Δ g q ≥ c 1 q 2 + a − T 2 q Re ⟨ ∂ T , ∂ q ⟩ g , a − T ∣ ∂ T ∣ g ≤ C q (168) for a constant c 1 > 0 c_1>0 c 1 > 0
Fix x ∈ C n x\in\mathbb{C}^n x ∈ C n B L ( x ) B_L(x) B L ( x ) ϕ = 1 − ∣ z − x ∣ 2 / L 2 \phi=1-\lvert z-x\rvert^2/L^2 ϕ = 1 − ∣ z − x ∣ 2 / L 2
∣ ∂ ϕ ∣ g 2 ≤ C / L 2 , Δ g ϕ ≥ − C / L 2 . (169) \lvert\partial\phi\rvert_g^2\le C/L^2,\qquad\Delta_g\phi\ge-C/L^2. \tag*{(169)} ∣ ∂ ϕ ∣ g 2 ≤ C / L 2 , Δ g ϕ ≥ − C / L 2 . (169) The function ϕ 2 q \phi^2q ϕ 2 q ∂ q = − 2 q ∂ ϕ / ϕ \partial q=-2q\,\partial\phi/\phi ∂ q = − 2 q ∂ ϕ / ϕ
0 ≥ Δ g ( ϕ 2 q ) ≥ c 1 ϕ 2 q 2 − 4 ϕ q a − T Re ⟨ ∂ T , ∂ ϕ ⟩ g + 2 ϕ q Δ g ϕ − 6 q ∣ ∂ ϕ ∣ g 2 ≥ c 1 2 ϕ 2 q 2 − C L 2 q . (170) \begin{aligned}
0\ge\Delta_g(\phi^2q)\ge c_1\phi^2q^2-\frac{4\phi q}{a-T}\operatorname{Re}\langle\partial T,\partial\phi\rangle_g+2\phi q\Delta_g\phi-6q\lvert\partial\phi\rvert_g^2 \\
&\ge\frac{c_1}{2}\phi^2q^2-\frac{C}{L^2}q. \tag*{(170)}
\end{aligned} 0 ≥ Δ g ( ϕ 2 q ) ≥ c 1 ϕ 2 q 2 − a − T 4 ϕq Re ⟨ ∂ T , ∂ ϕ ⟩ g + 2 ϕq Δ g ϕ − 6 q ∣ ∂ ϕ ∣ g 2 ≥ 2 c 1 ϕ 2 q 2 − L 2 C q . (170) In the last step, the drift term has size at most C ϕ q 3 / 2 ∣ ∂ ϕ ∣ g C\phi q^{3/2}\lvert\partial\phi\rvert_g C ϕ q 3/2 ∣ ∂ ϕ ∣ g q > 0 q>0 q > 0 max B L ( x ) ϕ 2 q ≤ C / L 2 \max_{B_L(x)}\phi^2q\le C/L^2 max B L ( x ) ϕ 2 q ≤ C / L 2 ϕ ( x ) = 1 \phi(x)=1 ϕ ( x ) = 1 q ( x ) ≤ C / L 2 q(x)\le C/L^2 q ( x ) ≤ C / L 2 L → ∞ L\to\infty L → ∞ q ( x ) = 0 q(x)=0 q ( x ) = 0 B = 0 B=0 B = 0 D g = 0 Dg=0 D g = 0
Recover the quadratic potential
A constant complex Hessian leaves a pluriharmonic part undetermined. This final step uses quadratic growth to show that this remaining part is also a polynomial of degree at most two, completing the smooth Liouville theorem. By Section A.3, the matrix g g g H = g H=g H = g v = u − H [ z ] v=u-H[z] v = u − H [ z ] D C 2 v = 0 D_{\mathbb{C}}^2v=0 D C 2 v = 0 v v v Δ R v = 0 \Delta_{\mathbb{R}}v=0 Δ R v = 0 Theorem A.1 give ∣ v ( z ) ∣ ≤ C ( 1 + ∣ z ∣ 2 ) |v(z)|\le C(1+|z|^2) ∣ v ( z ) ∣ ≤ C ( 1 + ∣ z ∣ 2 ) x x x
∣ D R 3 v ( x ) ∣ ≤ C n L 3 sup B L ( x ) ∣ v ∣ ≤ C x ( 1 + L 2 ) L 3 → 0. |D_{\mathbb{R}}^3v(x)|\le\frac{C_n}{L^3}\sup_{B_L(x)}|v|\le\frac{C_x(1+L^2)}{L^3}\to0. ∣ D R 3 v ( x ) ∣ ≤ L 3 C n B L ( x ) sup ∣ v ∣ ≤ L 3 C x ( 1 + L 2 ) → 0. Consequently D R 3 v = 0 D_{\mathbb{R}}^3v=0 D R 3 v = 0 v v v u u u 1 2 ξ T H 0 ξ \frac{1}{2}\xi^T H_0\xi 2 1 ξ T H 0 ξ Theorem A.1 , applied along each real ray, give 2 m I 2 n ≤ H 0 ≤ 2 M I 2 n 2mI_{2n}\le H_0\le2MI_{2n} 2 m I 2 n ≤ H 0 ≤ 2 M I 2 n Theorem A.1 .
A smooth counterexample under quadratic upper growth
Li and Wu [40 Remark 1.2] prove that a smooth entire solution of σ 2 ( D R 2 v ) = 1 \sigma_2(D_{\mathbb{R}}^2v)=1 σ 2 ( D R 2 v ) = 1 ∣ v ( x ) ∣ ≤ C ( 1 + ∣ x ∣ 2 ) |v(x)|\le C(1+|x|^2) ∣ v ( x ) ∣ ≤ C ( 1 + ∣ x ∣ 2 ) det D C 2 v = 1 \det D_{\mathbb{C}}^2v=1 det D C 2 v = 1 C 2 \mathbb{C}^2 C 2 σ k \sigma_k σ k C 2 \mathbb{C}^2 C 2
Proposition B.1 (Smooth counterexample). There exists a nonquadratic strictly plurisubharmonic function v ∈ C ∞ ( C 2 ) v\in C^\infty(\mathbb{C}^2) v ∈ C ∞ ( C 2 )
det D C 2 v = 1 , 0 < v ( z ) ≤ 8 ( 1 + ∣ z ∣ 2 ) . (171) \det D_{\mathbb{C}}^2v=1,\qquad0<v(z)\le8(1+|z|^2). \tag*{(171)} det D C 2 v = 1 , 0 < v ( z ) ≤ 8 ( 1 + ∣ z ∣ 2 ) . (171) Its complex Hessian is unbounded.
Proof. Construction and smoothness. Write x = Re z 1 x=\operatorname{Re} z_1 x = Re z 1 q = ∣ z 2 ∣ 2 q=|z_2|^2 q = ∣ z 2 ∣ 2 s > 0 s>0 s > 0 t ≥ 0 t\ge0 t ≥ 0
log s + s − q s = 2 x , t = q s . (172) \log s+s-\frac{q}{s}=2x,\qquad t=\frac{q}{s}. \tag*{(172)} log s + s − s q = 2 x , t = s q . (172) For fixed q ≥ 0 q\ge0 q ≥ 0 − ∞ -\infty − ∞ + ∞ +\infty + ∞ s s s
1 s + 1 + q s 2 = 1 + s + t s > 0. (173) \frac{1}{s}+1+\frac{q}{s^2}=\frac{1+s+t}{s}>0. \tag*{(173)} s 1 + 1 + s 2 q = s 1 + s + t > 0. (173) The monotonicity and endpoint limits above give a unique positive solution of (B.2). By (B.3), the implicit function theorem makes s s s ( x , q ) (x,q) ( x , q ) q = 0 q=0 q = 0 q q q
v = s + t + 1 2 ( s 2 + t 2 ) , K = 1 + s + t . (174) v=s+t+\frac{1}{2}(s^2+t^2),\qquad K=1+s+t. \tag*{(174)} v = s + t + 2 1 ( s 2 + t 2 ) , K = 1 + s + t . (174) Since s > 0 s>0 s > 0 t = q / s ≥ 0 t=q/s\ge0 t = q / s ≥ 0 C 2 \mathbb{C}^2 C 2 { z 2 = 0 } \{z_2=0\} { z 2 = 0 }
The complex Monge–Ampère equation. Differentiating (B.2) gives
s x = 2 s K , s q = 1 K , t x = − 2 t K , t q = 1 + s s K . (175) s_x=\frac{2s}{K},\qquad s_q=\frac{1}{K},\qquad t_x=-\frac{2t}{K},\qquad t_q=\frac{1+s}{sK}. \tag*{(175)} s x = K 2 s , s q = K 1 , t x = − K 2 t , t q = sK 1 + s . (175) Using (B.5) in (B.4), we obtain
v x = 2 ( s − t ) , v q = 1 + 1 s . (176) v_x=2(s-t),\qquad v_q=1+\frac{1}{s}. \tag*{(176)} v x = 2 ( s − t ) , v q = 1 + s 1 . (176) Differentiating (B.6) and using (B.5) once more gives
v x x = 4 ( s + t ) K , v x q = − 2 s K , v q q = − 1 s 2 K . (177) v_{xx}=\frac{4(s+t)}{K},\qquad v_{xq}=-\frac{2}{sK},\qquad v_{qq}=-\frac{1}{s^{2}K}. \tag*{(177)} v xx = K 4 ( s + t ) , v x q = − sK 2 , v q q = − s 2 K 1 . (177) By (B.2)–(B.4), v = v ( x , q ) v=v(x,q) v = v ( x , q ) x = Re z 1 x=\operatorname{Re}z_1 x = Re z 1 q = ∣ z 2 ∣ 2 q=|z_2|^2 q = ∣ z 2 ∣ 2
D C 2 v = ( 1 4 v x x 1 2 z ‾ 2 v x q 1 2 z 2 v x q v q + q v q q ) = ( s + t K − z 2 s K − z ‾ 2 s K 1 + 1 + s s K ) . (178) D_{\mathbb C}^{2}v=
\begin{pmatrix}
\frac{1}{4}v_{xx} & \frac{1}{2}\overline{z}_2v_{xq}\\
\frac{1}{2}z_2v_{xq} & v_q+qv_{qq}
\end{pmatrix}
=
\begin{pmatrix}
\frac{s+t}{K} & -\frac{z_2}{sK}\\
-\frac{\overline{z}_2}{sK} & 1+\frac{1+s}{sK}
\end{pmatrix}. \tag*{(178)} D C 2 v = ( 4 1 v xx 2 1 z 2 v x q 2 1 z 2 v x q v q + q v q q ) = ( K s + t − sK z 2 − sK z 2 1 + sK 1 + s ) . (178) By (B.2)–(B.4), q = s t q=st q = s t ( s + t ) ( 1 + s ) − t = s K (s+t)(1+s)-t=sK ( s + t ) ( 1 + s ) − t = sK
det D C 2 v = s + t K + ( s + t ) ( 1 + s ) − t s K 2 = s + t + 1 K = 1. (179) \det D_{\mathbb C}^{2}v=\frac{s+t}{K}+\frac{(s+t)(1+s)-t}{sK^2}=\frac{s+t+1}{K}=1. \tag*{(179)} det D C 2 v = K s + t + s K 2 ( s + t ) ( 1 + s ) − t = K s + t + 1 = 1. (179) The first diagonal entry in (B.8) is positive. Together with (B.9), this proves that D C 2 v D_{\mathbb C}^{2}v D C 2 v
Quadratic upper growth. When s ≥ 1 s\ge1 s ≥ 1
s 2 = q + 2 x s − s log s ≤ q + 2 ∣ x ∣ s ≤ q + 2 x 2 + 1 2 s 2 . s^2=q+2xs-s\log s\le q+2|x|s\le q+2x^2+\frac{1}{2}s^2. s 2 = q + 2 x s − s log s ≤ q + 2∣ x ∣ s ≤ q + 2 x 2 + 2 1 s 2 . Combining this with s 2 < 1 s^2<1 s 2 < 1 s < 1 s<1 s < 1
s 2 ≤ 1 + 2 q + 4 x 2 . (180) s^2\le1+2q+4x^2. \tag*{(180)} s 2 ≤ 1 + 2 q + 4 x 2 . (180) Since log s ≤ s \log s\le s log s ≤ s t ≤ 2 s + 2 ∣ x ∣ t\le2s+2|x| t ≤ 2 s + 2∣ x ∣ s t = q st=q s t = q
t 2 ≤ 2 q + 2 ∣ x ∣ t ≤ 2 q + 2 x 2 + 1 2 t 2 , hence t 2 ≤ 4 q + 4 x 2 . (181) t^2\le2q+2|x|t\le2q+2x^2+\frac{1}{2}t^2,\qquad\text{hence}\qquad t^2\le4q+4x^2. \tag*{(181)} t 2 ≤ 2 q + 2∣ x ∣ t ≤ 2 q + 2 x 2 + 2 1 t 2 , hence t 2 ≤ 4 q + 4 x 2 . (181) The bounds (B.10)–(B.11) and s + t ≤ 1 + 1 2 ( s 2 + t 2 ) s+t\le1+\frac{1}{2}(s^2+t^2) s + t ≤ 1 + 2 1 ( s 2 + t 2 )
0 < v ≤ 1 + s 2 + t 2 ≤ 2 + 6 q + 8 x 2 ≤ 8 ( 1 + ∣ z ∣ 2 ) . (182) 0<v\le1+s^2+t^2\le2+6q+8x^2\le8(1+|z|^2). \tag*{(182)} 0 < v ≤ 1 + s 2 + t 2 ≤ 2 + 6 q + 8 x 2 ≤ 8 ( 1 + ∣ z ∣ 2 ) . (182) Nonquadraticity. On { z 2 = 0 } \{z_2=0\} { z 2 = 0 }
log s + s = 2 x , D C 2 v = ( s 1 + s 0 0 1 + 1 s ) . (183) \log s+s=2x,\qquad D_{\mathbb C}^{2}v=
\begin{pmatrix}
\frac{s}{1+s} & 0\\
0 & 1+\frac{1}{s}
\end{pmatrix}. \tag*{(183)} log s + s = 2 x , D C 2 v = ( 1 + s s 0 0 1 + s 1 ) . (183) As x → − ∞ x\to-\infty x → − ∞ s → 0 s\to0 s → 0 v 2 2 ‾ → + ∞ v_{2\overline{2}}\to+\infty v 2 2 → + ∞ v v v
By (B.2)–(B.4), v v v Im z 1 \operatorname{Im}z_1 Im z 1 (4) , whose left-hand side tends to infinity along that direction for every α ∈ [ 0 , 1 ] \alpha\in[0,1] α ∈ [ 0 , 1 ] a = Re z 2 a=\operatorname{Re}z_2 a = Re z 2 z 2 z_2 z 2 q = s t q=st q = s t
v a a = 2 v q + 4 q v q q = 2 + 2 s − 4 t s K = − 2 13 when s = 1 2 , t = 5. v_{aa}=2v_q+4qv_{qq}=2+\frac{2}{s}-\frac{4t}{sK}=-\frac{2}{13}\qquad\text{when }s=\frac{1}{2},\ t=5. v aa = 2 v q + 4 q v q q = 2 + s 2 − sK 4 t = − 13 2 when s = 2 1 , t = 5. Such a point exists by (B.2), with z 2 = 5 / 2 z_2=\sqrt{5}/2 z 2 = 5 /2 x = ( log ( 1 / 2 ) + 1 / 2 − 5 ) / 2 x=(\log(1/2)+1/2-5)/2 x = ( log ( 1/2 ) + 1/2 − 5 ) /2
Verification of the continuous example
We verify the Bedford–Taylor equation asserted in Remark 1.7. Each summand in (15) is the modulus of a holomorphic function, so w w w { z 1 z 2 z 3 ≠ 0 } \{z_1z_2z_3 \ne0\} { z 1 z 2 z 3 = 0 } κ = 16 1 / 3 \kappa= 16^{1/3} κ = 1 6 1/3 r j = ∣ z j ∣ r_j = |z_j| r j = ∣ z j ∣ U = diag ( z 1 / r 1 , z 2 / r 2 , z 3 / r 3 ) U = \operatorname{diag}(z_1/r_1, z_2/r_2, z_3/r_3) U = diag ( z 1 / r 1 , z 2 / r 2 , z 3 / r 3 ) (15) gives
D C 2 w = κ 4 U ∗ N U , N = ( r 2 + r 3 r 1 1 1 1 r 1 + r 3 r 2 1 1 1 r 1 + r 2 r 3 ) . (184) D_{\mathbb{C}}^2w = \frac{\kappa}{4}U^*NU,\qquad N = \begin{pmatrix} \frac{r_2+r_3}{r_1} & 1 & 1 \\ 1 & \frac{r_1+r_3}{r_2} & 1 \\ 1 & 1 & \frac{r_1+r_2}{r_3} \end{pmatrix}. \tag*{(184)} D C 2 w = 4 κ U ∗ N U , N = r 1 r 2 + r 3 1 1 1 r 2 r 1 + r 3 1 1 1 r 3 r 1 + r 2 . (184) If d 1 , d 2 , d 3 d_1,d_2,d_3 d 1 , d 2 , d 3 N N N
det N = d 1 d 2 d 3 − d 1 − d 2 − d 3 + 2 = 4. \det N = d_1d_2d_3-d_1-d_2-d_3+2=4. det N = d 1 d 2 d 3 − d 1 − d 2 − d 3 + 2 = 4. Since U U U κ 3 = 16 \kappa^3=16 κ 3 = 16 det D C 2 w = 16 ⋅ 4 / 4 3 = 1 \det D_{\mathbb{C}}^2w=16\cdot4/4^3=1 det D C 2 w = 16 ⋅ 4/ 4 3 = 1 w w w [5 ] . The reference volume measure also gives them no mass. Hence ( − 1 ∂ ∂ ˉ w ) 3 = ω 0 3 (\sqrt{-1}\partial\bar\partial w)^3=\omega_0^3 ( − 1 ∂ ∂ ˉ w ) 3 = ω 0 3 C 3 \mathbb{C}^3 C 3
Acknowledgements
This research was assisted by ChatGPT.
