Introduction
One of the most studied nonlinear partial differential equations of parabolic type over the last six decades is the semilinear Cauchy problem
{ ∂ t u − Δ u = ∣ u ∣ b − 1 u in R n × ( 0 , T ) , u ( ⋅ , 0 ) = u 0 in R n , (1) \begin{cases}
\partial_t u-\Delta u=|u|^{b-1}u & \text{in }\mathbb{R}^n\times(0,T),\\
u(\cdot,0)=u_0 & \text{in }\mathbb{R}^n, \tag*{(1)}
\end{cases} { ∂ t u − Δ u = ∣ u ∣ b − 1 u u ( ⋅ , 0 ) = u 0 in R n × ( 0 , T ) , in R n , (1) where n ∈ N n\in\mathbb{N} n ∈ N b > 1 b>1 b > 1 0 < T ≤ ∞ 0<T\le\infty 0 < T ≤ ∞ (1) were obtained in [41 , 42 , 3 , 16 , 22 , 6 , 30 , 31 , 39 , 34 ] , etc. On the other hand, it follows from the earlier work of Fujita [15 ] that no nontrivial nonnegative global in time (T = ∞ T=\infty T = ∞ b < 1 + 2 / n b<1+2/n b < 1 + 2/ n b = 1 + 2 / n b=1+2/n b = 1 + 2/ n [17 ] and [41 , 42 ] ).
In this paper, we are interested in the Cauchy problem (1) with singular initial data u 0 u_0 u 0 L a L^a L a a ≥ 1 a\ge1 a ≥ 1 n ( b − 1 ) / 2 n(b-1)/2 n ( b − 1 ) /2 1 ≤ a < n ( b − 1 ) / 2 1\le a<n(b-1)/2 1 ≤ a < n ( b − 1 ) /2 u ∈ C ( [ 0 , T ] , L a ( R n ) ) u\in C([0,T],L^a(\mathbb{R}^n)) u ∈ C ([ 0 , T ] , L a ( R n )) u 0 ∈ L a ( R n ) u_0\in L^a(\mathbb{R}^n) u 0 ∈ L a ( R n ) [41 Theorem 1] ).
On the other hand, let b > 1 b>1 b > 1 a ≥ n ( b − 1 ) / 2 a\ge n(b-1)/2 a ≥ n ( b − 1 ) /2 a > 1 a>1 a > 1 a > n ( b − 1 ) / 2 a>n(b-1)/2 a > n ( b − 1 ) /2 a ≥ 1 a\ge1 a ≥ 1 [41 ] , Weissler used a fixed point theorem in a subspace of C ( [ 0 , T ] , L a ( R n ) ) C([0,T],L^a(\mathbb{R}^n)) C ([ 0 , T ] , L a ( R n )) u ∈ C ( [ 0 , T ] , L a ( R n ) ) ∩ L l o c ∞ ( ( 0 , T ] , L ∞ ( R n ) ) u\in C([0,T],L^a(\mathbb{R}^n))\cap L^\infty_{\mathrm{loc}}((0,T],L^\infty(\mathbb{R}^n)) u ∈ C ([ 0 , T ] , L a ( R n )) ∩ L loc ∞ (( 0 , T ] , L ∞ ( R n )) T = T ( u 0 ) > 0 T=T(u_0)>0 T = T ( u 0 ) > 0 (1) . Such solutions are sometimes referred to as Weissler’s solutions.
Later, Brezis and Cazenave [6 ] showed that this solution is unique in the class C ( [ 0 , T ] , L a ( R n ) ) ∩ L l o c ∞ ( ( 0 , T ] , L ∞ ( R n ) ) C([0,T],L^{a}(\mathbb{R}^{n})) \cap L^\infty_{\mathrm{loc}}((0,T],L^\infty(\mathbb{R}^{n})) C ([ 0 , T ] , L a ( R n )) ∩ L loc ∞ (( 0 , T ] , L ∞ ( R n )) a ≥ n ( b − 1 ) / 2 a \ge n(b-1)/2 a ≥ n ( b − 1 ) /2 a > b a>b a > b C ( [ 0 , T ] , L a ( R n ) ) C([0,T],L^{a}(\mathbb{R}^{n})) C ([ 0 , T ] , L a ( R n )) [41 ] for the case a > n ( b − 1 ) / 2 a>n(b-1)/2 a > n ( b − 1 ) /2 a > b a>b a > b [6 ] for the case a = n ( b − 1 ) / 2 a=n(b-1)/2 a = n ( b − 1 ) /2 a > b a>b a > b C ( [ 0 , T ] , L a ( R n ) ) C([0,T],L^{a}(\mathbb{R}^{n})) C ([ 0 , T ] , L a ( R n )) a = n ( b − 1 ) / 2 a=n(b-1)/2 a = n ( b − 1 ) /2 a = b a=b a = b a = b = n / ( n − 2 ) a=b=n/(n-2) a = b = n / ( n − 2 ) [40 ] and also [33 ] for the case of a bounded domain).
It is easy to verify that if u ( x , t ) u(x,t) u ( x , t ) (1) , then for any λ > 0 \lambda>0 λ > 0 u λ ( x , t ) : = λ 2 b − 1 u ( λ x , λ 2 t ) u_{\lambda}(x,t):=\lambda^{\frac{2}{b-1}}u(\lambda x,\lambda^{2}t) u λ ( x , t ) := λ b − 1 2 u ( λ x , λ 2 t ) R n × [ 0 , λ − 2 T ] \mathbb{R}^{n}\times[0,\lambda^{-2}T] R n × [ 0 , λ − 2 T ] u 0 λ ( x ) = λ 2 b − 1 u 0 ( λ x ) u_{0\lambda}(x)=\lambda^{\frac{2}{b-1}}u_{0}(\lambda x) u 0 λ ( x ) = λ b − 1 2 u 0 ( λ x ) b = 3 b=3 b = 3 [30 ] (see also [29 ] ), Y. Meyer treated the case b = n = 3 b=n=3 b = n = 3 (1) with b = n = 3 b=n=3 b = n = 3 [30 Conjecture 2.1] ).
Conjecture: There exists a Banach space ( Z , ∥ ⋅ ∥ Z ) (\mathcal{Z},\|\cdot\|_{\mathcal{Z}}) ( Z , ∥ ⋅ ∥ Z )
(i) ∥ f ∥ L 4 ( R 3 ) 2 ≲ ∥ ∇ f ∥ L 2 ( R 3 ) ∥ f ∥ Z \|f\|_{L^{4}(\mathbb{R}^{3})}^{2}\lesssim\|\nabla f\|_{L^{2}(\mathbb{R}^{3})}\|f\|_{\mathcal{Z}} ∥ f ∥ L 4 ( R 3 ) 2 ≲ ∥∇ f ∥ L 2 ( R 3 ) ∥ f ∥ Z
(ii) If u 0 ∈ C c ∞ ( R 3 ) u_{0}\in C_{c}^{\infty}(\mathbb{R}^{3}) u 0 ∈ C c ∞ ( R 3 ) ∥ u 0 ∥ Z ≤ η \|u_{0}\|_{\mathcal{Z}}\le\eta ∥ u 0 ∥ Z ≤ η T T T (1) with b = n = 3 b=n=3 b = n = 3 η \eta η
(iii) ∥ λ f ( λ ⋅ ) ∥ Z = ∥ f ∥ Z \|\lambda f(\lambda\cdot)\|_{\mathcal{Z}}=\|f\|_{\mathcal{Z}} ∥ λ f ( λ ⋅ ) ∥ Z = ∥ f ∥ Z λ > 0 \lambda>0 λ > 0
(iv) ∥ f ∥ Z \|f\|_{\mathcal{Z}} ∥ f ∥ Z f f f
Throughout the manuscript, for nonnegative numbers N , M N,M N , M N ≲ M N\lesssim M N ≲ M N ≤ C M N\le CM N ≤ C M C > 0 C>0 C > 0
Let B ˙ p , q α ( R n ) \dot{B}^{\alpha}_{p,q}(\mathbb{R}^{n}) B ˙ p , q α ( R n ) F ˙ p , q α ( R n ) \dot{F}^{\alpha}_{p,q}(\mathbb{R}^{n}) F ˙ p , q α ( R n ) α ∈ R \alpha\in\mathbb{R} α ∈ R p , q ∈ ( 0 , ∞ ] p,q\in(0,\infty] p , q ∈ ( 0 , ∞ ] R n \mathbb{R}^{n} R n § 2 ). Y. Meyer observed that this conjecture holds for Z = B ˙ 6 , ∞ − 1 / 2 ( R 3 ) \mathcal{Z}=\dot{B}^{-1/2}_{6,\infty}(\mathbb{R}^{3}) Z = B ˙ 6 , ∞ − 1/2 ( R 3 ) Z = B ˙ p , ∞ − 1 + 3 / p ( R 3 ) \mathcal{Z}=\dot{B}^{-1+3/p}_{p,\infty}(\mathbb{R}^{3}) Z = B ˙ p , ∞ − 1 + 3/ p ( R 3 ) 3 < p < 9 3<p<9 3 < p < 9 [5 , 22 , 31 ] ). Moreover, in these cases, property (ii) holds also for any u 0 ∈ L 3 ( R 3 ) u_{0}\in L^{3}(\mathbb{R}^{3}) u 0 ∈ L 3 ( R 3 ) L 3 ( R 3 ) ⊂ B ˙ p , ∞ − 1 + 3 / p ( R 3 ) ⊂ B ˙ ∞ , ∞ − 1 ( R 3 ) L^{3}(\mathbb{R}^{3})\subset\dot{B}^{-1+3/p}_{p,\infty}(\mathbb{R}^{3})\subset\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{3}) L 3 ( R 3 ) ⊂ B ˙ p , ∞ − 1 + 3/ p ( R 3 ) ⊂ B ˙ ∞ , ∞ − 1 ( R 3 ) 3 < p ≤ ∞ 3<p\le\infty 3 < p ≤ ∞ negative order lies in the fact that the condition of being sufficiently small in such spaces can be verified for highly oscillating data, which justifies property (iv). For example, if f ∈ L 3 ( R 3 ) f\in L^{3}(\mathbb{R}^{3}) f ∈ L 3 ( R 3 ) f k ( x ) = e i k ⋅ x f ( x ) f_{k}(x)=e^{ik\cdot x}f(x) f k ( x ) = e ik ⋅ x f ( x ) k ∈ Z 3 k\in\mathbb{Z}^{3} k ∈ Z 3 ∥ f k ∥ L 3 ( R 3 ) = ∥ f ∥ L 3 ( R 3 ) \|f_{k}\|_{L^{3}(\mathbb{R}^{3})}=\|f\|_{L^{3}(\mathbb{R}^{3})} ∥ f k ∥ L 3 ( R 3 ) = ∥ f ∥ L 3 ( R 3 ) ∥ f k ∥ B ˙ p , ∞ − 1 + 3 / p ( R 3 ) → 0 \|f_{k}\|_{\dot{B}^{-1+3/p}_{p,\infty}(\mathbb{R}^{3})}\to0 ∥ f k ∥ B ˙ p , ∞ − 1 + 3/ p ( R 3 ) → 0 ∣ k ∣ → ∞ |k|\to\infty ∣ k ∣ → ∞ 3 < p < ∞ 3<p<\infty 3 < p < ∞ [11 Section 4.1] ).
The main point of Meyer’s conjecture is to find the largest Banach space Z \mathcal{Z} Z [30 ] proved property (i) for Z = B ˙ ∞ , ∞ − 1 ( R 3 ) \mathcal{Z}=\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{3}) Z = B ˙ ∞ , ∞ − 1 ( R 3 ) B ˙ ∞ , ∞ − 1 ( R 3 ) \dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{3}) B ˙ ∞ , ∞ − 1 ( R 3 ) [5 Theorem 2.2] , where the authors showed that for any η > 0 \eta>0 η > 0 3 < q ≤ ∞ 3<q\leq\infty 3 < q ≤ ∞ u 0 ∈ S ( R 3 ) u_{0}\in\mathcal{S}(\mathbb{R}^{3}) u 0 ∈ S ( R 3 )
∥ u 0 ∥ B ˙ 9 , q − 2 / 3 ( R 3 ) ≤ η . \lVert u_{0}\rVert_{\dot{B}^{-2/3}_{9,q}(\mathbb{R}^{3})}\leq\eta. ∥ u 0 ∥ B ˙ 9 , q − 2/3 ( R 3 ) ≤ η . and the lifetime T T T Z = B ˙ 9 , q − 2 / 3 ( R 3 ) \mathcal{Z}=\dot{B}^{-2/3}_{9,q}(\mathbb{R}^{3}) Z = B ˙ 9 , q − 2/3 ( R 3 ) 3 < q ≤ ∞ 3<q\leq\infty 3 < q ≤ ∞ Z = B ˙ ∞ , ∞ − 1 ( R 3 ) \mathcal{Z}=\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{3}) Z = B ˙ ∞ , ∞ − 1 ( R 3 ) S ( R 3 ) \mathcal{S}(\mathbb{R}^{3}) S ( R 3 ) C c ∞ ( R 3 ) C^{\infty}_{c}(\mathbb{R}^{3}) C c ∞ ( R 3 ) [5 Theorem 2.2] for general n n n b ∈ N b\in\mathbb{N} b ∈ N b > 1 + 2 / n b>1+2/n b > 1 + 2/ n u 0 ∈ S ( R n ) u_{0}\in\mathcal{S}(\mathbb{R}^{n}) u 0 ∈ S ( R n ) ∥ u 0 ∥ B ˙ n b ( b − 1 ) / 2 , q − 2 / b \lVert u_{0}\rVert_{\dot{B}^{-2/b}_{n b(b-1)/2,q}} ∥ u 0 ∥ B ˙ nb ( b − 1 ) /2 , q − 2/ b q ∈ ( b , ∞ ] q\in(b,\infty] q ∈ ( b , ∞ ] q = b q=b q = b [5 page 12] : Let b > 1 + 2 / n b>1+2/n b > 1 + 2/ n u 0 u_{0} u 0 ∣ x ∣ → + ∞ |x|\to+\infty ∣ x ∣ → + ∞ ∥ u 0 ∥ B ˙ n b ( b − 1 ) / 2 , b − 2 / b ≪ 1 \lVert u_{0}\rVert_{\dot{B}^{-2/b}_{n b(b-1)/2,b}}\ll1 ∥ u 0 ∥ B ˙ nb ( b − 1 ) /2 , b − 2/ b ≪ 1 (1) is global in time?
One of the goals of this paper is to provide a positive answer to the above question, even for initial data in L a ( R n ) L^{a}(\mathbb{R}^{n}) L a ( R n ) a ≥ n ( b − 1 ) / 2 a\geq n(b-1)/2 a ≥ n ( b − 1 ) /2 (1) to have global in time solutions are included in our space. More precisely, our main results are stated in Theorems 4.9 and 4.11 below and our space of initial data is defined by
Z b , b ′ ( R n ) : = { f ∈ S ′ ( R n ) : ∥ f ∥ Z b , b ′ ( R n ) < + ∞ } . \mathcal{Z}_{b,b'}(\mathbb{R}^{n}):=\{f\in\mathcal{S}'(\mathbb{R}^{n}):\lVert f\rVert_{\mathcal{Z}_{b,b'}(\mathbb{R}^{n})}<+\infty\}. Z b , b ′ ( R n ) := { f ∈ S ′ ( R n ) : ∥ f ∥ Z b , b ′ ( R n ) < + ∞ } . where b > 1 + 2 / n b>1+2/n b > 1 + 2/ n b ′ = b / ( b − 1 ) b'=b/(b-1) b ′ = b / ( b − 1 )
∥ f ∥ Z b , b ′ ( R n ) : = sup E ⊂ R + n + 1 ( ∬ E ∣ t Δ f ( x ) ∣ b d x d t cap 2 , 1 , b ′ ( E ) ) 1 / b . \lVert f\rVert_{\mathcal{Z}_{b,b'}(\mathbb{R}^{n})}:=\sup_{E\subset\mathbb{R}^{n+1}_{+}}\left(\frac{\iint_{E}|t\Delta f(x)|^{b}\,dxdt}{\operatorname{cap}_{2,1,b'}(E)}\right)^{1/b}. ∥ f ∥ Z b , b ′ ( R n ) := E ⊂ R + n + 1 sup ( cap 2 , 1 , b ′ ( E ) ∬ E ∣ t Δ f ( x ) ∣ b d x d t ) 1/ b . Here the supremum is taken over all compact subsets E E E R + n + 1 = R n × ( 0 , ∞ ) \mathbb{R}^{n+1}_{+}=\mathbb{R}^{n}\times(0,\infty) R + n + 1 = R n × ( 0 , ∞ ) cap 2 , 1 , b ′ ( E ) \operatorname{cap}_{2,1,b'}(E) cap 2 , 1 , b ′ ( E )
cap 2 , 1 , b ′ ( E ) = inf Φ ∈ K ( E ) { ∫ R + n + 1 ( ∣ ∂ t Φ ∣ b ′ + ∣ D 2 Φ ∣ b ′ ) d x d t } . \operatorname{cap}_{2,1,b'}(E)=\inf_{\Phi\in K(E)}\left\{\int_{\mathbb{R}^{n+1}_{+}}\left(|\partial_{t}\Phi|^{b'}+|D^{2}\Phi|^{b'}\right)\,dxdt\right\}. cap 2 , 1 , b ′ ( E ) = Φ ∈ K ( E ) inf { ∫ R + n + 1 ( ∣ ∂ t Φ ∣ b ′ + ∣ D 2 Φ ∣ b ′ ) d x d t } . with
K ( E ) = { Φ ∈ C c ∞ ( R n + 1 ) , 0 ≤ Φ ≤ 1 , Φ = 1 in a neighborhood of E } . K(E)=\{\Phi\in C^{\infty}_{c}(\mathbb{R}^{n+1}),\ 0\leq\Phi\leq1,\ \Phi=1\text{ in a neighborhood of }E\}. K ( E ) = { Φ ∈ C c ∞ ( R n + 1 ) , 0 ≤ Φ ≤ 1 , Φ = 1 in a neighborhood of E } . See § 2 for different notions of capacities used in this paper. Of course, when dealing with Meyer’s conjecture, we take Z = Z 3 , 3 / 2 ( R 3 ) \mathcal{Z}=\mathcal{Z}_{3,3/2}(\mathbb{R}^{3}) Z = Z 3 , 3/2 ( R 3 ) Theorems 4.9 , 4.11 , and 3.3 below.
Theorem 1.1. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 u 0 ∈ S ′ ( R n ) u_0\in\mathcal{S}'(\mathbb{R}^n) u 0 ∈ S ′ ( R n ) ε = ε ( n , b ) > 0 \varepsilon=\varepsilon(n,b)>0 ε = ε ( n , b ) > 0 u 0 ∈ Z b , b ′ ( R n ) u_0\in Z_{b,b'}(\mathbb{R}^n) u 0 ∈ Z b , b ′ ( R n ) ∥ u 0 ∥ Z b , b ′ ( R n ) ≤ ε \lVert u_0\rVert_{Z_{b,b'}(\mathbb{R}^n)}\leq\varepsilon ∥ u 0 ∥ Z b , b ′ ( R n ) ≤ ε u u u (1) with T = ∞ T=\infty T = ∞ u ∈ L a ( R n ) u\in L^a(\mathbb{R}^n) u ∈ L a ( R n ) a ≥ n ( b − 1 ) 2 a\geq\frac{n(b-1)}{2} a ≥ 2 n ( b − 1 ) u ∈ C ( [ 0 , ∞ ) , L a ( R n ) ) u\in C([0,\infty),L^a(\mathbb{R}^n)) u ∈ C ([ 0 , ∞ ) , L a ( R n )) u u u
Conversely, if u u u R n \mathbb{R}^n R n (1) admits a nonnegative global in time solution u ∈ L l o c b ( R n × ( 0 , ∞ ) ) u\in L^b_{\mathrm{loc}}(\mathbb{R}^n\times(0,\infty)) u ∈ L loc b ( R n × ( 0 , ∞ )) u 0 ∈ Z b , b ′ ( R n ) u_0\in Z_{b,b'}(\mathbb{R}^n) u 0 ∈ Z b , b ′ ( R n )
Remark 1.2. Interestingly, one can show that Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^n) Z b , b ′ ( R n )
T b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) ↪ T b , 0 ( R n ) , ϵ > 0 , T_{b,\epsilon}(\mathbb{R}^n)\hookrightarrow Z_{b,b'}(\mathbb{R}^n)\hookrightarrow T_{b,0}(\mathbb{R}^n),\qquad\epsilon>0, T b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) ↪ T b , 0 ( R n ) , ϵ > 0 , where for b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 ϵ ≥ 0 \epsilon\geq0 ϵ ≥ 0
T b , ϵ ( R n ) : = { f ∈ S ′ ( R n ) : ∥ f ∥ T b , ϵ ( R n ) < + ∞ } T_{b,\epsilon}(\mathbb{R}^n):=\left\{f\in\mathcal{S}'(\mathbb{R}^n):\lVert f\rVert_{T_{b,\epsilon}(\mathbb{R}^n)}<+\infty\right\} T b , ϵ ( R n ) := { f ∈ S ′ ( R n ) : ∥ f ∥ T b , ϵ ( R n ) < + ∞ } with
∥ f ∥ T b , ϵ ( R n ) : = ( sup x ∈ R n , r > 0 r 2 b ′ / ( 1 + ϵ ) − n − 2 ∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ) 1 b ( 1 + ϵ ) . \lVert f\rVert_{T_{b,\epsilon}(\mathbb{R}^n)}:=\left(\sup_{x\in\mathbb{R}^n,\ r>0}r^{2b'/(1+\epsilon)-n-2}\int_0^{r^2}\int_{B_r(x)}\lvert e^{t\Delta}f\rvert^{b(1+\epsilon)}\,dy\,dt\right)^{\frac{1}{b(1+\epsilon)}}. ∥ f ∥ T b , ϵ ( R n ) := ( x ∈ R n , r > 0 sup r 2 b ′ / ( 1 + ϵ ) − n − 2 ∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ) b ( 1 + ϵ ) 1 . For this see Theorem 4.4 below. Note that the endpoint space T b , 0 ( R n ) T_{b,0}(\mathbb{R}^n) T b , 0 ( R n ) [21 ] ) in terms of scaling and optimal integrability. Indeed, in order to make sense of a solution u u u (1) we need at least that u ∈ L l o c b ( R n × [ 0 , ∞ ) ) u\in L^b_{\mathrm{loc}}(\mathbb{R}^n\times[0,\infty)) u ∈ L loc b ( R n × [ 0 , ∞ ))
sup x ∈ R n , r > 0 r 2 b ′ − n − 2 ∫ 0 r 2 ∫ B r ( x ) ∣ u ∣ b d y d t < + ∞ . \sup_{x\in\mathbb{R}^n,\ r>0}r^{2b'-n-2}\int_0^{r^2}\int_{B_r(x)}\lvert u\rvert^b\,dy\,dt<+\infty. x ∈ R n , r > 0 sup r 2 b ′ − n − 2 ∫ 0 r 2 ∫ B r ( x ) ∣ u ∣ b d y d t < + ∞. Thus, the space of initial data should at least be included in T b , 0 ( R n ) T_{b,0}(\mathbb{R}^n) T b , 0 ( R n )
Theorem 1.1 shows that the space Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^n) Z b , b ′ ( R n ) (1) at least when dealing with nonnegative initial data and nonnegative solutions. See Theorem 3.3 below for more results in the case of nonnegative initial data. In particular, it provides explicit characterizations of a (nonlinear) parabolic weighted norm inequality intrinsic to the Cauchy problem (1) discovered earlier in [3 ] (see also [20 ] ).
For initial tempered distributional data, as a consequence of Theorem 4.1 below, one has
B ˙ n b ( b − 1 ) / 2 , b − 2 / b ( R n ) ↪ F ˙ n b ( b − 1 ) / 2 , b − 2 / b ( R n ) ↪ Z b , b ′ ( R n ) , b > 1 + 2 n , \dot{B}^{-2/b}_{nb(b-1)/2,b}(\mathbb{R}^n)\hookrightarrow\dot{F}^{-2/b}_{nb(b-1)/2,b}(\mathbb{R}^n)\hookrightarrow Z_{b,b'}(\mathbb{R}^n),\qquad b>1+\frac{2}{n}, B ˙ nb ( b − 1 ) /2 , b − 2/ b ( R n ) ↪ F ˙ nb ( b − 1 ) /2 , b − 2/ b ( R n ) ↪ Z b , b ′ ( R n ) , b > 1 + n 2 , which provides the answer to the question of Brandolese and Cortez not only for the Besov space B ˙ n b ( b − 1 ) / 2 , b − 2 / b ( R n ) \dot{B}^{-2/b}_{nb(b-1)/2,b}(\mathbb{R}^n) B ˙ nb ( b − 1 ) /2 , b − 2/ b ( R n ) F ˙ n b ( b − 1 ) / 2 , b − 2 / b ( R n ) \dot{F}^{-2/b}_{nb(b-1)/2,b}(\mathbb{R}^n) F ˙ nb ( b − 1 ) /2 , b − 2/ b ( R n ) B ∞ n p − 2 b − 1 M q n q p B_{\infty}^{\frac{n}{p}-\frac{2}{b-1}}\mathcal{M}_{q}^{\frac{nq}{p}} B ∞ p n − b − 1 2 M q p n q b ≤ q ≤ p b \leq q \leq p b ≤ q ≤ p n ( b − 1 ) / 2 < p < n b ( b − 1 ) / 2 n(b-1)/2 < p < nb(b-1)/2 n ( b − 1 ) /2 < p < nb ( b − 1 ) /2 [22 ] (see § 2 for the definition of Besov-Morrey spaces). In Corollary 4.5 below, we show that in fact these Besov-Morrey spaces are also included in Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^{n}) Z b , b ′ ( R n )
The smallness of the norm of u 0 u_0 u 0 M b − 1 n q b − 1 ( R n ) \mathcal{M}_{b-1}^{\frac{nq}{b-1}}(\mathbb{R}^{n}) M b − 1 b − 1 n q ( R n ) 1 < q ≤ n ( b − 1 ) / 2 1 < q \leq n(b-1)/2 1 < q ≤ n ( b − 1 ) /2 (1) (see [39 ] ). More generally, Q.-H. Nguyen [34 ] obtained a similar existence result for u 0 u_0 u 0 μ \mu μ R n \mathbb{R}^{n} R n
∣ μ ∣ ( K ) ≤ A Cap 2 b , b ′ ( K ) |\mu|(K) \leq A\operatorname{Cap}_{\frac{2}{b},b'}(K) ∣ μ ∣ ( K ) ≤ A Cap b 2 , b ′ ( K ) for all compact sets K ⊂ R n K \subset\mathbb{R}^{n} K ⊂ R n Cap 2 b , b ′ ( ⋅ ) \operatorname{Cap}_{\frac{2}{b},b'}(\cdot) Cap b 2 , b ′ ( ⋅ ) 2 / b 2/b 2/ b § 2 ). These results are also special cases of our result in which one can take initial data in F ˙ b ′ − 2 b M ˙ b ′ 2 b ′ ( R n ) \dot{F}_{b'}^{-\frac{2}{b}}\dot{M}_{b'}^{\frac{2}{b'}}(\mathbb{R}^{n}) F ˙ b ′ − b 2 M ˙ b ′ b ′ 2 ( R n ) § 2 and Theorem 4.1 and Corollary 4.3).
Finally, we remark that, by Theorem 2.2 and Proposition 2.4 below, a tempered distribution f f f Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^{n}) Z b , b ′ ( R n ) b > 1 + 2 / n b > 1 + 2/n b > 1 + 2/ n C > 0 C > 0 C > 0
I 2 [ I 2 [ ν ] b ′ ] ≲ I 2 [ ν ] < + ∞ a.e. (2) \mathcal{I}_2[\mathcal{I}_2[\nu]^{b'}] \lesssim\mathcal{I}_2[\nu] < +\infty\quad\text{a.e.} \tag*{(2)} I 2 [ I 2 [ ν ] b ′ ] ≲ I 2 [ ν ] < + ∞ a.e. (2) where ν ( x , t ) = χ ( 0 , ∞ ) ( t ) ∣ e t Δ f ( x ) ∣ b \nu(x,t)=\chi_{(0,\infty)}(t)|e^{t\Delta}f(x)|^b ν ( x , t ) = χ ( 0 , ∞ ) ( t ) ∣ e t Δ f ( x ) ∣ b I 2 \mathcal{I}_2 I 2
I 2 [ ν ] ( x , t ) = ∫ R n + 1 ν ( y , s ) ( ∣ t − s ∣ + ∣ x − y ∣ 2 ) n / 2 d y d s , \mathcal{I}_2[\nu](x,t)=\int_{\mathbb{R}^{n+1}}\frac{\nu(y,s)}{(|t-s|+|x-y|^2)^{n/2}}\,dy\,ds, I 2 [ ν ] ( x , t ) = ∫ R n + 1 ( ∣ t − s ∣ + ∣ x − y ∣ 2 ) n /2 ν ( y , s ) d y d s , and χ I \chi_I χ I I I I
Then by the theory developed in [20 ] (see Theorem 4.9 and Proposition 2.9 of [20 ] ), condition (2) is also equivalent to a more abstract condition: There exist a function v ≥ ν v \geq\nu v ≥ ν R n + 1 \mathbb{R}^{n+1} R n + 1 B > 0 B > 0 B > 0 I 2 [ v b ′ ] ≤ B v \mathcal{I}_2[v^{b'}] \leq Bv I 2 [ v b ′ ] ≤ B v
Thus in the quadratic case b = 2 b=2 b = 2 Z 2 , 2 ( R n ) Z_{2,2}(\mathbb{R}^{n}) Z 2 , 2 ( R n ) n ≥ 3 n \geq3 n ≥ 3 V α , β V^{\alpha,\beta} V α , β α = 2 \alpha=2 α = 2 β = 0 \beta=0 β = 0 [24 ] to deal with the equation ∂ t u + ( − Δ ) α / 2 u = ( − Δ ) β / 2 u 2 \partial_tu+(-\Delta)^{\alpha/2}u=(-\Delta)^{\beta/2}u^2 ∂ t u + ( − Δ ) α /2 u = ( − Δ ) β /2 u 2 0 < α < n + 2 β 0 < \alpha< n+2\beta 0 < α < n + 2 β 0 < β < α 0 < \beta< \alpha 0 < β < α
Capacities and function spaces
Parabolic capacities and potentials
In what follows, for a Borel set E ⊂ R d E \subset\mathbb{R}^{d} E ⊂ R d d ∈ N d \in\mathbb{N} d ∈ N M ( E ) \mathcal{M}(E) M ( E ) E E E M + ( E ) \mathcal{M}^{+}(E) M + ( E )
Given μ ∈ M + ( R n ) \mu\in\mathcal{M}^{+}(\mathbb{R}^{n}) μ ∈ M + ( R n ) n ∈ N n \in\mathbb{N} n ∈ N α ∈ ( 0 , n ) \alpha\in(0,n) α ∈ ( 0 , n )
I α [ μ ] ( x ) = ∫ R n 1 ∣ x − y ∣ n − α d μ ( y ) , x ∈ R n . \mathcal{I}_{\alpha}[\mu](x)=\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n-\alpha}}\,d\mu(y),\quad x\in\mathbb{R}^{n}. I α [ μ ] ( x ) = ∫ R n ∣ x − y ∣ n − α 1 d μ ( y ) , x ∈ R n . Similarly, for ν ∈ M + ( R n + 1 ) \nu\in\mathcal{M}^{+}(\mathbb{R}^{n+1}) ν ∈ M + ( R n + 1 ) β ∈ ( 0 , n + 2 ) \beta\in(0,n+2) β ∈ ( 0 , n + 2 )
I β [ ν ] ( x , t ) = ∫ R n + 1 1 ( ∣ t − s ∣ + ∣ x − y ∣ 2 ) ( n + 2 − β ) / 2 d ν ( y , s ) , \mathbb{I}_{\beta}[\nu](x,t)=\int_{\mathbb{R}^{n+1}}\frac{1}{\left(|t-s|+|x-y|^{2}\right)^{(n+2-\beta)/2}}\,d\nu(y,s), I β [ ν ] ( x , t ) = ∫ R n + 1 ( ∣ t − s ∣ + ∣ x − y ∣ 2 ) ( n + 2 − β ) /2 1 d ν ( y , s ) , where x ∈ R n x \in\mathbb{R}^{n} x ∈ R n t ∈ R t \in\mathbb{R} t ∈ R
Denote by B r ( x ) B_r(x) B r ( x ) R n \mathbb{R}^{n} R n x x x r > 0 r>0 r > 0 Q r ( x , t ) = B r ( x ) × ( t − r 2 / 2 , t + r 2 / 2 ) Q_r(x,t)=B_r(x)\times(t-r^2/2,t+r^2/2) Q r ( x , t ) = B r ( x ) × ( t − r 2 /2 , t + r 2 /2 )
I α [ μ ] ( x ) = ( n − α ) ∫ 0 ∞ μ ( B r ( x ) ) r n − α d r r , x ∈ R n , \mathbb{I}_{\alpha}[\mu](x)=(n-\alpha)\int_{0}^{\infty}\frac{\mu(B_r(x))}{r^{n-\alpha}}\frac{dr}{r},\qquad x\in\mathbb{R}^{n}, I α [ μ ] ( x ) = ( n − α ) ∫ 0 ∞ r n − α μ ( B r ( x )) r d r , x ∈ R n , and
I β [ ν ] ( x , t ) ≃ ∫ 0 ∞ ν ( Q r ( x , t ) ) r n + 2 − β d r r , ( x , t ) ∈ R n + 1 . \mathbb{I}_{\beta}[\nu](x,t)\simeq\int_{0}^{\infty}\frac{\nu(Q_r(x,t))}{r^{n+2-\beta}}\frac{dr}{r},\qquad(x,t)\in\mathbb{R}^{n+1}. I β [ ν ] ( x , t ) ≃ ∫ 0 ∞ r n + 2 − β ν ( Q r ( x , t )) r d r , ( x , t ) ∈ R n + 1 . Recall that the heat kernel on R n + 1 \mathbb{R}^{n+1} R n + 1
H ( x , t ) = 1 ( 4 π t ) n / 2 e − ∣ x ∣ 2 4 t χ ( 0 , ∞ ) ( t ) . H(x,t)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x|^2}{4t}}\chi_{(0,\infty)}(t). H ( x , t ) = ( 4 π t ) n /2 1 e − 4 t ∣ x ∣ 2 χ ( 0 , ∞ ) ( t ) . We also define the heat potential of a measure ν ∈ M + ( R n + 1 ) \nu\in\mathcal{M}^{+}(\mathbb{R}^{n+1}) ν ∈ M + ( R n + 1 )
H [ ν ] ( x , t ) = [ H ∗ ν ] ( x , t ) = ∫ R n + 1 H ( x − y , t − s ) d ν ( y , s ) H[\nu](x,t)=[H\ast\nu](x,t)=\int_{\mathbb{R}^{n+1}}H(x-y,t-s)d\nu(y,s) H [ ν ] ( x , t ) = [ H ∗ ν ] ( x , t ) = ∫ R n + 1 H ( x − y , t − s ) d ν ( y , s ) for any ( x , t ) ∈ R n + 1 (x,t)\in\mathbb{R}^{n+1} ( x , t ) ∈ R n + 1
With these, we can now define, respectively, the Riesz, the parabolic Riesz, and the heat capacities as follows:
Cap I α , p ( K ) = inf { ∥ f ∥ L p ( R n ) p : f ≥ 0 , I α [ f ] ≥ 1 on K } \operatorname{Cap}_{\mathbb{I}_{\alpha},p}(K)=\inf\left\{\|f\|_{L^{p}(\mathbb{R}^{n})}^{p}: f\geq0,\ \mathbb{I}_{\alpha}[f]\geq1\ \text{on }K\right\} Cap I α , p ( K ) = inf { ∥ f ∥ L p ( R n ) p : f ≥ 0 , I α [ f ] ≥ 1 on K } for any set K ⊂ R n K\subset\mathbb{R}^{n} K ⊂ R n α ∈ ( 0 , n ) \alpha\in(0,n) α ∈ ( 0 , n ) 1 < p < n α 1<p<\frac{n}{\alpha} 1 < p < α n
Cap I β , p 1 ( E ) = inf { ∥ f ∥ L p 1 ( R n + 1 ) p 1 : f ≥ 0 , I β [ f ] ≥ 1 on E } , \operatorname{Cap}_{\mathbb{I}_{\beta},p_1}(E)=\inf\left\{\|f\|_{L^{p_1}(\mathbb{R}^{n+1})}^{p_1}: f\geq0,\ \mathbb{I}_{\beta}[f]\geq1\ \text{on }E\right\}, Cap I β , p 1 ( E ) = inf { ∥ f ∥ L p 1 ( R n + 1 ) p 1 : f ≥ 0 , I β [ f ] ≥ 1 on E } , Cap H , p 2 ( E ) = inf { ∥ f ∥ L p 2 ( R n + 1 ) p 2 : f ≥ 0 , H [ f ] ≥ 1 on E } \operatorname{Cap}_{H,p_2}(E)=\inf\left\{\|f\|_{L^{p_2}(\mathbb{R}^{n+1})}^{p_2}: f\geq0,\ H[f]\geq1\ \text{on }E\right\} Cap H , p 2 ( E ) = inf { ∥ f ∥ L p 2 ( R n + 1 ) p 2 : f ≥ 0 , H [ f ] ≥ 1 on E } for any set E ⊂ R n + 1 E\subset\mathbb{R}^{n+1} E ⊂ R n + 1 β ∈ ( 0 , n + 2 ) \beta\in(0,n+2) β ∈ ( 0 , n + 2 ) 1 < p 1 < n + 2 β 1<p_1<\frac{n+2}{\beta} 1 < p 1 < β n + 2 1 < p 2 < n + 2 2 1<p_2<\frac{n+2}{2} 1 < p 2 < 2 n + 2
We next introduce the following variational homogeneous parabolic capacities. For p > 1 p>1 p > 1 E ⊂ R n + 1 E\subset\mathbb{R}^{n+1} E ⊂ R n + 1
cap 2 , 1 , p ( E ) = inf { ∫ R n + 1 ( ∣ ∂ t Φ ∣ p + ∣ D 2 Φ ∣ p ) d x d t } , \operatorname{cap}_{2,1,p}(E)=\inf\left\{\int_{\mathbb{R}^{n+1}}\left(|\partial_t\Phi|^p+|D^2\Phi|^p\right)dxdt\right\}, cap 2 , 1 , p ( E ) = inf { ∫ R n + 1 ( ∣ ∂ t Φ ∣ p + ∣ D 2 Φ ∣ p ) d x d t } , where the infimum is taken over all functions Φ ∈ C c ∞ ( R n + 1 ) \Phi\in C^\infty_c(\mathbb{R}^{n+1}) Φ ∈ C c ∞ ( R n + 1 ) 0 ≤ Φ ≤ 1 0\leq\Phi\leq1 0 ≤ Φ ≤ 1 Φ ≥ 1 \Phi\geq1 Φ ≥ 1 E E E Φ = 1 \Phi=1 Φ = 1 E E E
c 2 , 1 , p ( E ) = inf { ∫ R n + 1 ( ∣ ∂ t Φ ∣ p + ∣ D 2 Φ ∣ p ) d x d t } , c_{2,1,p}(E)=\inf\left\{\int_{\mathbb{R}^{n+1}}\left(|\partial_t\Phi|^p+|D^2\Phi|^p\right)dxdt\right\}, c 2 , 1 , p ( E ) = inf { ∫ R n + 1 ( ∣ ∂ t Φ ∣ p + ∣ D 2 Φ ∣ p ) d x d t } , where the infimum is taken over all Φ ∈ C c ∞ ( R n + 1 ) \Phi\in C^\infty_c(\mathbb{R}^{n+1}) Φ ∈ C c ∞ ( R n + 1 ) Φ ≥ 1 \Phi\geq1 Φ ≥ 1 E E E
We will need the following parabolic Hedberg type lemma.
Lemma 2.1. Let 0 < α < β < n + 2 0 < \alpha< \beta< n+2 0 < α < β < n + 2 μ ∈ M + ( R n + 1 ) \mu\in\mathcal{M}^{+}(\mathbb{R}^{n+1}) μ ∈ M + ( R n + 1 )
I α [ μ ] ( x , t ) ≤ c M β μ ( x , t ) α / β M μ ( x , t ) 1 − α / β ≤ c I β [ μ ] ( x , t ) α / β M μ ( x , t ) 1 − α / β I_{\alpha}[\mu](x,t) \le c M_{\beta}\mu(x,t)^{\alpha/\beta}M\mu(x,t)^{1-\alpha/\beta} \le c I_{\beta}[\mu](x,t)^{\alpha/\beta}M\mu(x,t)^{1-\alpha/\beta} I α [ μ ] ( x , t ) ≤ c M β μ ( x , t ) α / β M μ ( x , t ) 1 − α / β ≤ c I β [ μ ] ( x , t ) α / β M μ ( x , t ) 1 − α / β for all ( x , t ) ∈ R n + 1 (x,t) \in\mathbb{R}^{n+1} ( x , t ) ∈ R n + 1
M β μ ( x , t ) = sup r > 0 μ ( Q r ( x , t ) ) r n + 2 − β , 0 ≤ β < n + 2 , M_{\beta}\mu(x,t)=\sup_{r>0}\frac{\mu(Q_r(x,t))}{r^{n+2-\beta}}, \qquad0\le\beta<n+2, M β μ ( x , t ) = r > 0 sup r n + 2 − β μ ( Q r ( x , t )) , 0 ≤ β < n + 2 , and M μ ( x , t ) = M 0 μ ( x , t ) M\mu(x,t)=M_0\mu(x,t) M μ ( x , t ) = M 0 μ ( x , t ) μ \mu μ
Proof. It is enough to prove the first inequality. For any δ > 0 \delta>0 δ > 0
I α [ μ ] ( x , t ) ≲ ∫ 0 δ μ ( Q r ( x , t ) ) r n + 2 − α d r r + ∫ δ ∞ μ ( Q r ( x , t ) ) r n + 2 − α d r r I_{\alpha}[\mu](x,t) \lesssim\int_0^\delta\frac{\mu(Q_r(x,t))}{r^{n+2-\alpha}}\frac{dr}{r}+\int_\delta^\infty\frac{\mu(Q_r(x,t))}{r^{n+2-\alpha}}\frac{dr}{r} I α [ μ ] ( x , t ) ≲ ∫ 0 δ r n + 2 − α μ ( Q r ( x , t )) r d r + ∫ δ ∞ r n + 2 − α μ ( Q r ( x , t )) r d r ≲ M μ ( x , t ) δ α + M β [ μ ] ( x , t ) δ α − β . \lesssim M\mu(x,t)\delta^\alpha+M_\beta[\mu](x,t)\delta^{\alpha-\beta}. ≲ M μ ( x , t ) δ α + M β [ μ ] ( x , t ) δ α − β . We may assume that both M μ ( x , t ) M\mu(x,t) M μ ( x , t ) M β [ μ ] ( x , t ) M_\beta[\mu](x,t) M β [ μ ] ( x , t )
δ = [ M β [ μ ] ( x , t ) M μ ( x , t ) ] 1 / β , \delta=\left[\frac{M_\beta[\mu](x,t)}{M\mu(x,t)}\right]^{1/\beta}, δ = [ M μ ( x , t ) M β [ μ ] ( x , t ) ] 1/ β , we obtain the lemma. □
The following equivalence of parabolic capacities will be important for our development later.
Theorem 2.2. Let 1 < p < n + 2 2 1<p<\frac{n+2}{2} 1 < p < 2 n + 2 C 1 , C 2 , C 3 , C_1,C_2,C_3, C 1 , C 2 , C 3 , C 4 C_4 C 4
c 2 , 1 , p ( E ) ≤ C 1 cap 2 , 1 , p ( E ) ≤ C 2 Cap I 2 , p ( E ) c_{2,1,p}(E)\le C_1\operatorname{cap}_{2,1,p}(E)\le C_2\operatorname{Cap}_{\mathbb{I}_2,p}(E) c 2 , 1 , p ( E ) ≤ C 1 cap 2 , 1 , p ( E ) ≤ C 2 Cap I 2 , p ( E ) ≤ C 3 Cap H , p ( E ) ≤ C 4 c 2 , 1 , p ( E ) (3) \le C_3\operatorname{Cap}_{H,p}(E)\le C_4c_{2,1,p}(E) \tag*{(3)} ≤ C 3 Cap H , p ( E ) ≤ C 4 c 2 , 1 , p ( E ) (3) for all compact sets E ⊂ R n + 1 E\subset\mathbb{R}^{n+1} E ⊂ R n + 1
Proof. The first inequality in (3) is trivial and the bound
Cap I 2 , p ( E ) ≤ c Cap H , p ( E ) \operatorname{Cap}_{\mathbb{I}_2,p}(E)\le c\operatorname{Cap}_{H,p}(E) Cap I 2 , p ( E ) ≤ c Cap H , p ( E ) is also obvious since H ( x , t ) ≤ c ( ∣ t ∣ + ∣ x ∣ 2 ) − n / 2 H(x,t)\le c\left(|t|+|x|^2\right)^{-n/2} H ( x , t ) ≤ c ( ∣ t ∣ + ∣ x ∣ 2 ) − n /2 R n + 1 ∖ ( 0 , 0 ) \mathbb{R}^{n+1}\setminus(0,0) R n + 1 ∖ ( 0 , 0 ) Φ ∈ C 0 ∞ ( R n + 1 ) \Phi\in C^\infty_0(\mathbb{R}^{n+1}) Φ ∈ C 0 ∞ ( R n + 1 ) Φ ≥ 1 \Phi\ge1 Φ ≥ 1 E ⊂ R n + 1 E\subset\mathbb{R}^{n+1} E ⊂ R n + 1
Φ = H ∗ ( ∂ t Φ − Δ Φ ) ≤ H ∗ ( ∣ ∂ t Φ − Δ Φ ∣ ) , \Phi=H*(\partial_t\Phi-\Delta\Phi)\le H*(|\partial_t\Phi-\Delta\Phi|), Φ = H ∗ ( ∂ t Φ − ΔΦ ) ≤ H ∗ ( ∣ ∂ t Φ − ΔΦ∣ ) , and thus
Cap H , p ( E ) ≤ ∫ R n + 1 ∣ ∂ t Φ − Δ Φ ∣ p d x d t ≲ ∫ R n + 1 ( ∣ ∂ t Φ ∣ p + ∣ D 2 Φ ∣ p ) d x d t . \operatorname{Cap}_{H,p}(E)\le\int_{\mathbb{R}^{n+1}}|\partial_t\Phi-\Delta\Phi|^pdxdt\lesssim\int_{\mathbb{R}^{n+1}}(|\partial_t\Phi|^p+|D^2\Phi|^p)dxdt. Cap H , p ( E ) ≤ ∫ R n + 1 ∣ ∂ t Φ − ΔΦ ∣ p d x d t ≲ ∫ R n + 1 ( ∣ ∂ t Φ ∣ p + ∣ D 2 Φ ∣ p ) d x d t . Minimizing the right-hand side above over all such Φ \Phi Φ
Cap H , p ( E ) ≲ c 2 , 1 , p ( E ) . \operatorname{Cap}_{H,p}(E)\lesssim c_{2,1,p}(E). Cap H , p ( E ) ≲ c 2 , 1 , p ( E ) . It remains to show the second inequality in (3) . Since Cap Π 2 , p \operatorname{Cap}_{\Pi_2,p} Cap Π 2 , p
cap 2 , 1 , p ( E ) ≲ Cap Π 2 , p ( O ) (4) \operatorname{cap}_{2,1,p}(E) \lesssim\operatorname{Cap}_{\Pi_2,p}(O) \tag*{(4)} cap 2 , 1 , p ( E ) ≲ Cap Π 2 , p ( O ) (4) for any bounded open set O ⊃ E O \supset E O ⊃ E c c c E E E O O O [1 ] for the elliptic case; see also [25 ] ).
Let f ∈ L p ( R n + 1 ) f \in L^p(\mathbb{R}^{n+1}) f ∈ L p ( R n + 1 ) f ≥ 0 f \ge0 f ≥ 0 I 2 [ f ] ≥ 1 \mathcal{I}_2[f] \ge1 I 2 [ f ] ≥ 1 O O O O O O E E E 1 < p < n + 2 2 1 < p < \frac{n+2}{2} 1 < p < 2 n + 2
∫ R n + 1 ∖ Q R ( 0 ) f ( y , s ) ( ∣ t − s ∣ + ∣ x − y ∣ 2 ) n / 2 d y d s ≤ 1 2 \int_{\mathbb{R}^{n+1}\setminus Q_R(0)} \frac{f(y,s)}{\left(|t-s|+|x-y|^2\right)^{n/2}}\,dy\,ds \le\frac{1}{2} ∫ R n + 1 ∖ Q R ( 0 ) ( ∣ t − s ∣ + ∣ x − y ∣ 2 ) n /2 f ( y , s ) d y d s ≤ 2 1 for all ( x , t ) ∈ O (x,t) \in O ( x , t ) ∈ O R > 0 R>0 R > 0 f f f 2 f χ Q R ( 0 ) 2f\chi_{Q_R(0)} 2 f χ Q R ( 0 ) R R R f f f
Let B r ( z ) B_r(z) B r ( z ) z z z r > 0 r>0 r > 0 R n + 1 \mathbb{R}^{n+1} R n + 1 φ \varphi φ R n + 1 \mathbb{R}^{n+1} R n + 1 φ ∈ C ∞ ( B 1 ( 0 ) ) \varphi\in C^\infty(B_1(0)) φ ∈ C ∞ ( B 1 ( 0 )) φ \varphi φ ∫ R n + 1 φ d x d t = 1 \int_{\mathbb{R}^{n+1}}\varphi\,dx\,dt=1 ∫ R n + 1 φ d x d t = 1 ϵ > 0 \epsilon>0 ϵ > 0 φ ϵ ( x , t ) = 1 ϵ n + 1 φ ( x / ϵ , t / ϵ ) \varphi_\epsilon(x,t)=\frac{1}{\epsilon^{n+1}}\varphi(x/\epsilon,t/\epsilon) φ ϵ ( x , t ) = ϵ n + 1 1 φ ( x / ϵ , t / ϵ ) ϵ \epsilon ϵ
φ ϵ ∗ I 2 [ f ] = I 2 [ φ ϵ ∗ f ] ≥ 1 \varphi_\epsilon* \mathcal{I}_2[f]=\mathcal{I}_2[\varphi_\epsilon*f]\ge1 φ ϵ ∗ I 2 [ f ] = I 2 [ φ ϵ ∗ f ] ≥ 1 in O ′ O' O ′ O ′ O' O ′ E ⊂ O ′ ⊂ O ′ ‾ ⊂ O E\subset O'\subset\overline{O'}\subset O E ⊂ O ′ ⊂ O ′ ⊂ O
Let T ∈ C ∞ ( R , [ 0 , ∞ ) ) T\in C^\infty(\mathbb{R},[0,\infty)) T ∈ C ∞ ( R , [ 0 , ∞ )) T ( s ) = 0 T(s)=0 T ( s ) = 0 s ≤ 1 2 s\le\frac{1}{2} s ≤ 2 1 T ( s ) = 1 T(s)=1 T ( s ) = 1 s ≥ 1 s\ge1 s ≥ 1
Φ ϵ = T ( I 2 [ φ ϵ ∗ f ] ) . \Phi_\epsilon=T(\mathcal{I}_2[\varphi_\epsilon*f]). Φ ϵ = T ( I 2 [ φ ϵ ∗ f ]) . Then we have
∂ t Φ ϵ = T ′ ( I 2 [ φ ϵ ∗ f ] ) ∂ t I 2 [ φ ϵ ∗ f ] \partial_t\Phi_\epsilon=T'(\mathcal{I}_2[\varphi_\epsilon*f])\partial_t\mathcal{I}_2[\varphi_\epsilon*f] ∂ t Φ ϵ = T ′ ( I 2 [ φ ϵ ∗ f ]) ∂ t I 2 [ φ ϵ ∗ f ] and for any i , j ∈ { 1 , 2 , … , n } i,j\in\{1,2,\ldots,n\} i , j ∈ { 1 , 2 , … , n }
D i j Φ ϵ = T ′ ′ ( I 2 [ φ ϵ ∗ f ] ) D i I 2 [ φ ϵ ∗ f ] D j I 2 [ φ ϵ ∗ f ] + T ′ ( I 2 [ φ ϵ ∗ f ] ) D i j I 2 [ φ ϵ ∗ f ] . D_{ij}\Phi_\epsilon=T''(\mathcal{I}_2[\varphi_\epsilon*f])D_i\mathcal{I}_2[\varphi_\epsilon*f]D_j\mathcal{I}_2[\varphi_\epsilon*f]+T'(\mathcal{I}_2[\varphi_\epsilon*f])D_{ij}\mathcal{I}_2[\varphi_\epsilon*f]. D ij Φ ϵ = T ′′ ( I 2 [ φ ϵ ∗ f ]) D i I 2 [ φ ϵ ∗ f ] D j I 2 [ φ ϵ ∗ f ] + T ′ ( I 2 [ φ ϵ ∗ f ]) D ij I 2 [ φ ϵ ∗ f ] . Thus it follows from Lemma 2.1 that
∣ ∂ t Φ ϵ ∣ + ∣ D i j Φ ϵ ∣ ≲ ∣ ∂ t I 2 [ φ ϵ ∗ f ] ∣ + ∣ D i j I 2 [ φ ϵ ∗ f ] ∣ + I 2 [ φ ϵ ∗ f ] I 1 [ φ ϵ ∗ f ] − 1 ∣ I 1 [ φ ϵ ∗ f ] ∣ 2 |\partial_t\Phi_\epsilon|+|D_{ij}\Phi_\epsilon|\lesssim|\partial_t\mathcal{I}_2[\varphi_\epsilon*f]|+|D_{ij}\mathcal{I}_2[\varphi_\epsilon*f]|+\mathcal{I}_2[\varphi_\epsilon*f]\mathcal{I}_1[\varphi_\epsilon*f]^{-1}|\mathcal{I}_1[\varphi_\epsilon*f]|^2 ∣ ∂ t Φ ϵ ∣ + ∣ D ij Φ ϵ ∣ ≲ ∣ ∂ t I 2 [ φ ϵ ∗ f ] ∣ + ∣ D ij I 2 [ φ ϵ ∗ f ] ∣ + I 2 [ φ ϵ ∗ f ] I 1 [ φ ϵ ∗ f ] − 1 ∣ I 1 [ φ ϵ ∗ f ] ∣ 2 ≲ ∣ ∂ t I 2 [ φ ϵ ∗ f ] ∣ + ∣ D i j I 2 [ φ ϵ ∗ f ] ∣ + M ( φ ϵ ∗ f ) , \lesssim|\partial_t\mathcal{I}_2[\varphi_\epsilon*f]|+|D_{ij}\mathcal{I}_2[\varphi_\epsilon*f]|+M(\varphi_\epsilon*f), ≲ ∣ ∂ t I 2 [ φ ϵ ∗ f ] ∣ + ∣ D ij I 2 [ φ ϵ ∗ f ] ∣ + M ( φ ϵ ∗ f ) , where in the first inequality we used the fact that s T ′ ′ ( s ) ≤ c sT''(s)\le c s T ′′ ( s ) ≤ c s ≥ 0 s\ge0 s ≥ 0 ∣ D i I 2 [ φ ϵ ∗ f ] ∣ ≲ ∣ I 1 [ φ ϵ ∗ f ] ∣ |D_i\mathcal{I}_2[\varphi_\epsilon*f]|\lesssim|\mathcal{I}_1[\varphi_\epsilon*f]| ∣ D i I 2 [ φ ϵ ∗ f ] ∣ ≲ ∣ I 1 [ φ ϵ ∗ f ] ∣ i ∈ { 1 , 2 , … , n } i\in\{1,2,\ldots,n\} i ∈ { 1 , 2 , … , n }
∂ t I 2 [ φ ϵ ∗ f ] = K 0 ∗ ( φ ϵ ∗ f ) and D i j I 2 [ φ ϵ ∗ f ] = K 1 ∗ ( φ ϵ ∗ f ) \partial_t\mathcal{I}_2[\varphi_\epsilon*f]=K_0*(\varphi_\epsilon*f)\qquad\text{and}\qquad D_{ij}\mathcal{I}_2[\varphi_\epsilon*f]=K_1*(\varphi_\epsilon*f) ∂ t I 2 [ φ ϵ ∗ f ] = K 0 ∗ ( φ ϵ ∗ f ) and D ij I 2 [ φ ϵ ∗ f ] = K 1 ∗ ( φ ϵ ∗ f ) are Calderón–Zygmund operators acting on φ ϵ ∗ f \varphi_\epsilon*f φ ϵ ∗ f K 0 K_0 K 0 K 1 K_1 K 1 M M M L p ( R n + 1 ) L^p(\mathbb{R}^{n+1}) L p ( R n + 1 )
∫ R n + 1 ( ∣ ∂ t Φ ϵ ∣ p + ∣ D i j Φ ϵ ∣ p ) d x d t ≲ ∫ R n + 1 ( φ ϵ ∗ f ) p d x d t ≲ ∫ R n + 1 f p d x d t \int_{\mathbb{R}^{n+1}}\left(|\partial_t\Phi_\epsilon|^p+|D_{ij}\Phi_\epsilon|^p\right)\,dx\,dt\lesssim\int_{\mathbb{R}^{n+1}}(\varphi_\epsilon*f)^p\,dx\,dt\lesssim\int_{\mathbb{R}^{n+1}}f^p\,dx\,dt ∫ R n + 1 ( ∣ ∂ t Φ ϵ ∣ p + ∣ D ij Φ ϵ ∣ p ) d x d t ≲ ∫ R n + 1 ( φ ϵ ∗ f ) p d x d t ≲ ∫ R n + 1 f p d x d t Note that 0 ≤ Φ ε ≤ 1 0 \le\Phi_{\varepsilon} \le1 0 ≤ Φ ε ≤ 1 Φ ε = 1 \Phi_{\varepsilon}=1 Φ ε = 1 O ′ ⊃ E O' \supset E O ′ ⊃ E Φ ε ∈ C c ∞ ( R n + 1 ) \Phi_{\varepsilon} \in C^\infty_c(\mathbb{R}^{n+1}) Φ ε ∈ C c ∞ ( R n + 1 ) f f f
cap 2 , 1 , p ( E ) ≤ ∫ R n + 1 ∣ ∂ t Φ ε ∣ p + ∣ D 2 Φ ε ∣ p d x d t ≲ ∫ R n + 1 f p d x d t . \operatorname{cap}_{2,1,p}(E) \le\int_{\mathbb{R}^{n+1}} \left|\partial_t\Phi_{\varepsilon}\right|^p+\left|D^2\Phi_{\varepsilon}\right|^p\,dxdt \lesssim\int_{\mathbb{R}^{n+1}} f^p\,dxdt. cap 2 , 1 , p ( E ) ≤ ∫ R n + 1 ∣ ∂ t Φ ε ∣ p + D 2 Φ ε p d x d t ≲ ∫ R n + 1 f p d x d t . Minimizing the right-hand side over all such f f f (4) as desired. □
We record the following important characterizations of parabolic trace inequality that can be found in [34 ] , Theorem 4.36; see also [27 ] for the elliptic case.
Proposition 2.4. Let ν ∈ M + ( R n + 1 ) \nu\in\mathcal{M}^+(\mathbb{R}^{n+1}) ν ∈ M + ( R n + 1 ) 1 < p < n + 2 β 1<p<\frac{n+2}{\beta} 1 < p < β n + 2
∫ R n + 1 I β [ f ] p d ν ≤ C 1 ∫ R n + 1 f p d x d t \int_{\mathbb{R}^{n+1}} I_\beta[f]^p\,d\nu\le C_1\int_{\mathbb{R}^{n+1}} f^p\,dxdt ∫ R n + 1 I β [ f ] p d ν ≤ C 1 ∫ R n + 1 f p d x d t ν ( E ) ≤ C 2 Cap I β , p ( E ) . \nu(E)\le C_2\operatorname{Cap}_{I_{\beta},p}(E). ν ( E ) ≤ C 2 Cap I β , p ( E ) . ∫ E I β [ ν ] p ′ d x d t ≤ C 3 Cap I β , p ( E ) , p ′ = p p − 1 . \int_E I_\beta[\nu]^{p'}\,dxdt\le C_3\operatorname{Cap}_{I_{\beta},p}(E),\qquad p'=\frac{p}{p-1}. ∫ E I β [ ν ] p ′ d x d t ≤ C 3 Cap I β , p ( E ) , p ′ = p − 1 p . I β [ ( I β [ ν ] ) p ′ ] ≤ C 4 I β [ ν ] a.e. I_\beta\left[(I_\beta[\nu])^{p'}\right]\le C_4I_\beta[\nu]\qquad\text{a.e.} I β [ ( I β [ ν ] ) p ′ ] ≤ C 4 I β [ ν ] a.e. Here C 1 C_1 C 1 C 2 C_2 C 2 C 3 C_3 C 3 C 4 C_4 C 4 ν \nu ν E E E
C 1 ≃ C 2 ≃ ( C 3 ) p − 1 p ≃ ( C 4 ) p − 1 . C_1\simeq C_2\simeq(C_3)^{\frac{p-1}{p}}\simeq(C_4)^{p-1}. C 1 ≃ C 2 ≃ ( C 3 ) p p − 1 ≃ ( C 4 ) p − 1 . Triebel-Lizorkin and Besov type spaces
Let χ ∈ C c ∞ ( R n ) \chi\in C^\infty_c(\mathbb{R}^n) χ ∈ C c ∞ ( R n ) 0 ≤ χ ≤ 1 0 \leq\chi\leq1 0 ≤ χ ≤ 1 χ ( ξ ) = 1 \chi(\xi)=1 χ ( ξ ) = 1 ∣ ξ ∣ ≤ 1 |\xi|\leq1 ∣ ξ ∣ ≤ 1 χ ( ξ ) = 0 \chi(\xi)=0 χ ( ξ ) = 0 ∣ ξ ∣ ≥ 2 |\xi|\geq2 ∣ ξ ∣ ≥ 2 j ∈ Z j\in\mathbb{Z} j ∈ Z
φ j ( ξ ) = φ ( 2 − j ξ ) , where φ ( ξ ) = χ ( ξ ) − χ ( 2 ξ ) , (5) \varphi_j(\xi)=\varphi(2^{-j}\xi), \quad\text{where } \varphi(\xi)=\chi(\xi)-\chi(2\xi), \tag*{(5)} φ j ( ξ ) = φ ( 2 − j ξ ) , where φ ( ξ ) = χ ( ξ ) − χ ( 2 ξ ) , (5) so that
∑ j ∈ Z φ j ( ξ ) = 1 , ∀ ξ ≠ 0. \sum_{j\in\mathbb{Z}}\varphi_j(\xi)=1,\quad\forall\xi\ne0. j ∈ Z ∑ φ j ( ξ ) = 1 , ∀ ξ = 0. The homogeneous Littlewood-Paley projection is the Fourier multiplier operator
Δ ˙ j f = φ j ( D ) f = F − 1 ( φ ( 2 − j ⋅ ) F f ) = 2 n j f ∗ F − 1 ( φ ) ( 2 j ⋅ ) . \dot{\Delta}_j f=\varphi_j(D)f=\mathcal{F}^{-1}\bigl(\varphi(2^{-j}\cdot)\mathcal{F}f\bigr)=2^{nj}f\ast\mathcal{F}^{-1}(\varphi)(2^j\cdot). Δ ˙ j f = φ j ( D ) f = F − 1 ( φ ( 2 − j ⋅ ) F f ) = 2 nj f ∗ F − 1 ( φ ) ( 2 j ⋅ ) . defined for f ∈ S ′ ( R n ) f\in\mathcal{S}'(\mathbb{R}^n) f ∈ S ′ ( R n ) F \mathcal{F} F F − 1 \mathcal{F}^{-1} F − 1
F h ( ξ ) : = ∫ R n h ( x ) e − 2 π i x ⋅ ξ d x , ∀ h ∈ S ( R n ) . \mathcal{F}h(\xi):=\int_{\mathbb{R}^n}h(x)e^{-2\pi ix\cdot\xi}\,dx,\quad\forall h\in\mathcal{S}(\mathbb{R}^n). F h ( ξ ) := ∫ R n h ( x ) e − 2 π i x ⋅ ξ d x , ∀ h ∈ S ( R n ) . Note that Δ ˙ j f = Δ ˙ j ( f + P ) \dot{\Delta}_j f=\dot{\Delta}_j(f+P) Δ ˙ j f = Δ ˙ j ( f + P ) P P P R n \mathbb{R}^n R n
f = ∑ j ∈ Z Δ ˙ j f in S ′ ( R n ) / P , f=\sum_{j\in\mathbb{Z}}\dot{\Delta}_j f\quad\text{in }\mathcal{S}'(\mathbb{R}^n)/P, f = j ∈ Z ∑ Δ ˙ j f in S ′ ( R n ) / P , where S ′ ( R n ) / P \mathcal{S}'(\mathbb{R}^n)/P S ′ ( R n ) / P R n \mathbb{R}^n R n [23 Proposition 3.1] ). On the other hand, for any N ∈ Z N\in\mathbb{Z} N ∈ Z f ∈ S ′ ( R n ) f\in\mathcal{S}'(\mathbb{R}^n) f ∈ S ′ ( R n )
f = F − 1 ( χ ( 2 − N ⋅ ) F f ) + ∑ j = N ∞ Δ ˙ j f in S ′ ( R n ) . f=\mathcal{F}^{-1}\bigl(\chi(2^{-N}\cdot)\mathcal{F}f\bigr)+\sum_{j=N}^{\infty}\dot{\Delta}_j f\quad\text{in }\mathcal{S}'(\mathbb{R}^n). f = F − 1 ( χ ( 2 − N ⋅ ) F f ) + j = N ∑ ∞ Δ ˙ j f in S ′ ( R n ) . Moreover, if lim N → − ∞ F − 1 ( χ ( 2 − N ⋅ ) F f ) = 0 \lim_{N\to-\infty}\mathcal{F}^{-1}\bigl(\chi(2^{-N}\cdot)\mathcal{F}f\bigr)=0 lim N → − ∞ F − 1 ( χ ( 2 − N ⋅ ) F f ) = 0 S ′ ( R n ) \mathcal{S}'(\mathbb{R}^n) S ′ ( R n ) [23 Theorem 3.1] ) that
f = ∑ j ∈ Z Δ ˙ j f in S ′ ( R n ) . f=\sum_{j\in\mathbb{Z}}\dot{\Delta}_j f\quad\text{in }\mathcal{S}'(\mathbb{R}^n). f = j ∈ Z ∑ Δ ˙ j f in S ′ ( R n ) . Definition 2.5. Let s ∈ R s\in\mathbb{R} s ∈ R p , q ∈ ( 0 , ∞ ] p,q\in(0,\infty] p , q ∈ ( 0 , ∞ ]
∥ f ∥ F ˙ p , q s ( R n ) : = ∥ ( ∑ j ∈ Z [ 2 j s Δ ˙ j f ] q ) 1 / q ∥ L p , q ∈ ( 0 , ∞ ] , p ∈ ( 0 , ∞ ) 1 , \|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^n)}:=\left\|\left(\sum_{j\in\mathbb{Z}}\left[2^{js}\dot{\Delta}_j f\right]^q\right)^{1/q}\right\|_{L^p},\quad q\in(0,\infty],\ p\in(0,\infty)^{1}, ∥ f ∥ F ˙ p , q s ( R n ) := j ∈ Z ∑ [ 2 j s Δ ˙ j f ] q 1/ q L p , q ∈ ( 0 , ∞ ] , p ∈ ( 0 , ∞ ) 1 , ∥ f ∥ B ˙ p , q s ( R n ) : = ( ∑ j ∈ Z [ 2 j s ∥ Δ ˙ j f ∥ L p ] q ) 1 / q , q ∈ ( 0 , ∞ ] , p ∈ ( 0 , ∞ ] , \|f\|_{\dot{B}^{s}_{p,q}(\mathbb{R}^n)}:=\left(\sum_{j\in\mathbb{Z}}\left[2^{js}\|\dot{\Delta}_j f\|_{L^p}\right]^q\right)^{1/q},\quad q\in(0,\infty],\ p\in(0,\infty], ∥ f ∥ B ˙ p , q s ( R n ) := j ∈ Z ∑ [ 2 j s ∥ Δ ˙ j f ∥ L p ] q 1/ q , q ∈ ( 0 , ∞ ] , p ∈ ( 0 , ∞ ] , where in the case q = ∞ q=\infty q = ∞ ℓ q ( Z ) \ell^q(\mathbb{Z}) ℓ q ( Z ) ℓ ∞ ( Z ) \ell^\infty(\mathbb{Z}) ℓ ∞ ( Z ) F ˙ p , q s ( R n ) \dot{F}^{s}_{p,q}(\mathbb{R}^n) F ˙ p , q s ( R n ) B ˙ p , q s ( R n ) \dot{B}^{s}_{p,q}(\mathbb{R}^{n}) B ˙ p , q s ( R n ) f ∈ S ′ ( R n ) / P ( R n ) f \in\mathcal{S}'(\mathbb{R}^{n})/\mathcal{P}(\mathbb{R}^{n}) f ∈ S ′ ( R n ) / P ( R n ) ∥ f ∥ F ˙ p , q s < + ∞ \|f\|_{\dot{F}^{s}_{p,q}} < +\infty ∥ f ∥ F ˙ p , q s < + ∞ ∥ f ∥ B ˙ p , q s < + ∞ \|f\|_{\dot{B}^{s}_{p,q}} < +\infty ∥ f ∥ B ˙ p , q s < + ∞
It is easy to see from the definition that
B ˙ p , min ( p , q ) s ( R n ) ↪ F ˙ p , q s ( R n ) ↪ B ˙ p , max ( p , q ) s ( R n ) . (6) \dot{B}^{s}_{p,\min(p,q)}(\mathbb{R}^{n}) \hookrightarrow\dot{F}^{s}_{p,q}(\mathbb{R}^{n}) \hookrightarrow\dot{B}^{s}_{p,\max(p,q)}(\mathbb{R}^{n}). \tag*{(6)} B ˙ p , m i n ( p , q ) s ( R n ) ↪ F ˙ p , q s ( R n ) ↪ B ˙ p , m a x ( p , q ) s ( R n ) . (6) More generally, we introduce the following definition.
Definition 2.6. Let s ∈ R s \in\mathbb{R} s ∈ R q ∈ ( 0 , ∞ ] q \in(0,\infty] q ∈ ( 0 , ∞ ] X X X R n \mathbb{R}^{n} R n ∥ ⋅ ∥ X \|\cdot\|_{X} ∥ ⋅ ∥ X
∥ f ∥ F ˙ q s X ( R n ) : = ∥ { ∑ j ∈ Z ( 2 j s Δ ˙ j f ) q } 1 / q ∥ X , q ∈ ( 0 , ∞ ] , \|f\|_{\dot{F}^{s}_{q}X(\mathbb{R}^{n})}:=\left\|\left\{\sum_{j\in\mathbb{Z}}(2^{js}\dot{\Delta}_{j}f)^{q}\right\}^{1/q}\right\|_{X},\qquad q\in(0,\infty], ∥ f ∥ F ˙ q s X ( R n ) := ⎩ ⎨ ⎧ j ∈ Z ∑ ( 2 j s Δ ˙ j f ) q ⎭ ⎬ ⎫ 1/ q X , q ∈ ( 0 , ∞ ] , ∥ f ∥ B ˙ q s X ( R n ) : = { ∑ j ∈ Z ( 2 j s ∥ Δ ˙ j f ∥ X ) q } 1 / q , q ∈ ( 0 , ∞ ] . \|f\|_{\dot{B}^{s}_{q}X(\mathbb{R}^{n})}:=\left\{\sum_{j\in\mathbb{Z}}\left(2^{js}\|\dot{\Delta}_{j}f\|_{X}\right)^{q}\right\}^{1/q},\qquad q\in(0,\infty]. ∥ f ∥ B ˙ q s X ( R n ) := ⎩ ⎨ ⎧ j ∈ Z ∑ ( 2 j s ∥ Δ ˙ j f ∥ X ) q ⎭ ⎬ ⎫ 1/ q , q ∈ ( 0 , ∞ ] . With this, we set
F ˙ q s X ( R n ) : = { f ∈ S ′ ( R n ) / P ( R n ) : ∥ f ∥ F ˙ q s X ( R n ) < + ∞ } \dot{F}^{s}_{q}X(\mathbb{R}^{n}):=\{f\in\mathcal{S}'(\mathbb{R}^{n})/\mathcal{P}(\mathbb{R}^{n}):\|f\|_{\dot{F}^{s}_{q}X(\mathbb{R}^{n})}<+\infty\} F ˙ q s X ( R n ) := { f ∈ S ′ ( R n ) / P ( R n ) : ∥ f ∥ F ˙ q s X ( R n ) < + ∞ } and
B ˙ q s X ( R n ) : = { f ∈ S ′ ( R n ) / P ( R n ) : ∥ f ∥ B ˙ q s X ( R n ) < + ∞ } . \dot{B}^{s}_{q}X(\mathbb{R}^{n}):=\{f\in\mathcal{S}'(\mathbb{R}^{n})/\mathcal{P}(\mathbb{R}^{n}):\|f\|_{\dot{B}^{s}_{q}X(\mathbb{R}^{n})}<+\infty\}. B ˙ q s X ( R n ) := { f ∈ S ′ ( R n ) / P ( R n ) : ∥ f ∥ B ˙ q s X ( R n ) < + ∞ } . In applications, one of our choices will be the weighted Lebesgue space X = L s ( w ) X=L^{s}(w) X = L s ( w ) s ∈ ( 0 , ∞ ) s\in(0,\infty) s ∈ ( 0 , ∞ ) w w w A ∞ A_{\infty} A ∞ R n \mathbb{R}^{n} R n w w w A ∞ A_{\infty} A ∞ w w w A p A_{p} A p p > 1 p>1 p > 1
[ w ] A p : = ( sup B ⊂ R n 1 ∣ B ∣ ∫ B w d y ) ( sup B 1 ∣ B ∣ ∫ B w 1 1 − p d y ) p − 1 < + ∞ [w]_{A_{p}}:=\left(\sup_{B\subset\mathbb{R}^{n}}\frac{1}{|B|}\int_{B}w\,dy\right)\left(\sup_{B}\frac{1}{|B|}\int_{B}w^{\frac{1}{1-p}}\,dy\right)^{p-1}<+\infty [ w ] A p := ( B ⊂ R n sup ∣ B ∣ 1 ∫ B w d y ) ( B sup ∣ B ∣ 1 ∫ B w 1 − p 1 d y ) p − 1 < + ∞ for some p > 1 p>1 p > 1 A p A_{p} A p p > 1 p>1 p > 1 A 1 A_{1} A 1 w ∈ L l o c 1 ( R n ) w\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n}) w ∈ L loc 1 ( R n )
sup r > 0 1 ∣ B r ( x ) ∣ ∫ B r ( x ) w ( y ) d y ≤ C w ( x ) \sup_{r>0}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}w(y)\,dy\leq Cw(x) r > 0 sup ∣ B r ( x ) ∣ 1 ∫ B r ( x ) w ( y ) d y ≤ C w ( x ) for a.e. x ∈ R n x\in\mathbb{R}^{n} x ∈ R n C C C [ w ] A 1 [w]_{A_{1}} [ w ] A 1 A 1 A_{1} A 1 w w w
Another example is the Sobolev multiplier type space X = M p α , r ( R n ) X=M^{\alpha,r}_{p}(\mathbb{R}^{n}) X = M p α , r ( R n )
Definition 2.7. Let p > 0 p>0 p > 0 α > 0 \alpha>0 α > 0 r > 1 r>1 r > 1 α r < n \alpha r<n α r < n M p α , r ( R n ) M^{\alpha,r}_{p}(\mathbb{R}^{n}) M p α , r ( R n ) f ∈ L l o c p ( R n ) f\in L^{p}_{\mathrm{loc}}(\mathbb{R}^{n}) f ∈ L loc p ( R n )
( ∫ R n I α [ h ] ∣ f ∣ p d x ) 1 / p ≤ C ( ∫ R n h r d x ) 1 / p (7) \left(\int_{\mathbb{R}^{n}}I_{\alpha}[h]|f|^{p}\,dx\right)^{1/p}\leq C\left(\int_{\mathbb{R}^{n}}h^{r}\,dx\right)^{1/p} \tag*{(7)} ( ∫ R n I α [ h ] ∣ f ∣ p d x ) 1/ p ≤ C ( ∫ R n h r d x ) 1/ p (7) holds for all h ∈ L r ( R n ) h\in L^{r}(\mathbb{R}^{n}) h ∈ L r ( R n ) h ≥ 0 h\geq0 h ≥ 0 M p α , r ( R n ) M^{\alpha,r}_{p}(\mathbb{R}^{n}) M p α , r ( R n ) C C C (7) .
In practice, it is not easy to work with (7) . The following characterizations for the quasi-norm of M p α , r ( R n ) \mathcal{M}^{\alpha,r}_{p}(\mathbb{R}^{n}) M p α , r ( R n )
Proposition 2.8. Let p > 0 p>0 p > 0 α > 0 \alpha>0 α > 0 r > 1 r>1 r > 1 α r < n \alpha r<n α r < n
∥ f ∥ M p α , r ( R n ) ≃ sup K ( ∫ K ∣ f ( x ) ∣ p d x Cap I α , r ( K ) ) 1 / p , (8) \|f\|_{\mathcal{M}^{\alpha,r}_{p}(\mathbb{R}^{n})}\simeq\sup_{K}\left(\frac{\int_{K}|f(x)|^{p}\,dx}{\operatorname{Cap}_{I_{\alpha,r}}(K)}\right)^{1/p}, \tag*{(8)} ∥ f ∥ M p α , r ( R n ) ≃ K sup ( Cap I α , r ( K ) ∫ K ∣ f ( x ) ∣ p d x ) 1/ p , (8) where the supremum is taken over all compact sets K ⊂ R n K\subset\mathbb{R}^{n} K ⊂ R n
∥ f ∥ M p α , r ( R n ) ≃ sup w ( ∫ R n ∣ f ∣ p w d x ) 1 / p , (9) \|f\|_{\mathcal{M}^{\alpha,r}_{p}(\mathbb{R}^{n})}\simeq\sup_{w}\left(\int_{\mathbb{R}^{n}}|f|^{p}w\,dx\right)^{1/p}, \tag*{(9)} ∥ f ∥ M p α , r ( R n ) ≃ w sup ( ∫ R n ∣ f ∣ p w d x ) 1/ p , (9) where the supremum is taken over all A 1 A_{1} A 1 w w w R n \mathbb{R}^{n} R n w w w Cap I α , r \operatorname{Cap}_{I_{\alpha,r}} Cap I α , r
∫ 0 ∞ Cap I α , r ( { w > t } ) d t ≤ 1 , \int_{0}^{\infty}\operatorname{Cap}_{I_{\alpha,r}}(\{w>t\})\,dt\leq1, ∫ 0 ∞ Cap I α , r ({ w > t }) d t ≤ 1 , and [ w ] A 1 ≤ c ‾ ( n , α , r ) [w]_{A_{1}}\leq\overline{c}(n,\alpha,r) [ w ] A 1 ≤ c ( n , α , r ) c ‾ ( n , α , r ) > 0 \overline{c}(n,\alpha,r)>0 c ( n , α , r ) > 0
Recall that a function v : R n → [ − ∞ , ∞ ] v:\mathbb{R}^{n}\to[-\infty,\infty] v : R n → [ − ∞ , ∞ ] Cap I α , r \operatorname{Cap}_{I_{\alpha,r}} Cap I α , r ϵ > 0 \epsilon>0 ϵ > 0 O O O Cap I α , r ( O ) < ϵ \operatorname{Cap}_{I_{\alpha,r}}(O)<\epsilon Cap I α , r ( O ) < ϵ v v v R n ∖ O \mathbb{R}^{n}\setminus O R n ∖ O
For the equivalence in (8) , see [26 , 2 ] . On the other hand, the equivalence (9) was obtained in [36 Theorem 9.2] . The constant c ‾ \overline{c} c c ‾ = [ ∣ ⋅ ∣ − n + α ] A 1 \overline{c}=[|\cdot|^{-n+\alpha}]_{A_{1}} c = [ ∣ ⋅ ∣ − n + α ] A 1 r r r [37 Theorem 6.7] ). Both [36 ] and [37 ] considered only the case p > 1 p>1 p > 1 p > 0 p>0 p > 0
The second embedding in (11) follows from the fact that Cap α , r ( B R ) ≃ R n − α r \operatorname{Cap}_{\alpha,r}(B_R)\simeq R^{n-\alpha r} Cap α , r ( B R ) ≃ R n − α r [13 ] and [38 ] .
By (6) , (10) , and (11) , we can easily deduce that, for 0 < ϵ < n α r − 1 0<\epsilon<\frac{n}{\alpha r}-1 0 < ϵ < α r n − 1
B ˙ q , min ( n p α r , q ) s ( R n ) ↪ F ˙ q s L ∞ n p α r ( R n ) ↪ F ˙ q s M p ( 1 + ϵ ) , α r ( 1 + ϵ ) ( R n ) , \dot{B}^{s}_{q,\min\left(\frac{np}{\alpha r},q\right)}(\mathbb{R}^{n})\hookrightarrow\dot{F}^{s}_{q}L^{\frac{np}{\alpha r}}_{\infty}(\mathbb{R}^{n})\hookrightarrow\dot{F}^{s}_{q}\mathcal{M}^{p(1+\epsilon),\alpha r(1+\epsilon)}(\mathbb{R}^{n}), B ˙ q , m i n ( α r n p , q ) s ( R n ) ↪ F ˙ q s L ∞ α r n p ( R n ) ↪ F ˙ q s M p ( 1 + ϵ ) , α r ( 1 + ϵ ) ( R n ) , and
F ˙ q s M p ( 1 + ϵ ) , α r ( 1 + ϵ ) ( R n ) ↪ F ˙ q s M p α , r ( R n ) ↪ F ˙ q s M p , α r ( R n ) \dot{F}^{s}_{q}\mathcal{M}^{p(1+\epsilon),\alpha r(1+\epsilon)}(\mathbb{R}^{n})\hookrightarrow\dot{F}^{s}_{q}\mathcal{M}^{\alpha,r}_{p}(\mathbb{R}^{n})\hookrightarrow\dot{F}^{s}_{q}\mathcal{M}^{p,\alpha r}(\mathbb{R}^{n}) F ˙ q s M p ( 1 + ϵ ) , α r ( 1 + ϵ ) ( R n ) ↪ F ˙ q s M p α , r ( R n ) ↪ F ˙ q s M p , α r ( R n ) for any s ∈ R s\in\mathbb{R} s ∈ R 0 < q ≤ ∞ 0<q\leq\infty 0 < q ≤ ∞
On the other hand, F ˙ q s M p α , r ( R n ) \dot{F}^{s}_{q}\mathcal{M}^{\alpha,r}_{p}(\mathbb{R}^{n}) F ˙ q s M p α , r ( R n )
Lemma 2.10. Let ( s , α ) ∈ R × R + (s,\alpha)\in\mathbb{R}\times\mathbb{R}_{+} ( s , α ) ∈ R × R + ( p , q , r ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ] × ( 1 , ∞ ) (p,q,r)\in(0,\infty)\times(0,\infty]\times(1,\infty) ( p , q , r ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ] × ( 1 , ∞ ) α r < n \alpha r<n α r < n
F ˙ q s M p α , r ( R n ) ↪ B ˙ ∞ , ∞ s − α r p ( R n ) . \dot{F}^{s}_{q}\mathcal{M}^{\alpha,r}_{p}(\mathbb{R}^{n})\hookrightarrow\dot{B}^{s-\frac{\alpha r}{p}}_{\infty,\infty}(\mathbb{R}^{n}). F ˙ q s M p α , r ( R n ) ↪ B ˙ ∞ , ∞ s − p α r ( R n ) . Proof. Since
F ˙ q s M p α , r ( R n ) ↪ F ˙ ∞ s M p , α r ( R n ) ↪ B ˙ ∞ s M p , α r ( R n ) , \dot{F}^{s}_{q}\mathcal{M}^{\alpha,r}_{p}(\mathbb{R}^{n})\hookrightarrow\dot{F}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^{n})\hookrightarrow\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^{n}), F ˙ q s M p α , r ( R n ) ↪ F ˙ ∞ s M p , α r ( R n ) ↪ B ˙ ∞ s M p , α r ( R n ) , it suffices to show that
B ˙ ∞ s M p , α r ( R n ) ↪ B ˙ ∞ , ∞ s − α r p ( R n ) . (12) \dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^{n})\hookrightarrow\dot{B}^{s-\frac{\alpha r}{p}}_{\infty,\infty}(\mathbb{R}^{n}). \tag*{(12)} B ˙ ∞ s M p , α r ( R n ) ↪ B ˙ ∞ , ∞ s − p α r ( R n ) . (12) Let f ∈ B ˙ ∞ s M p , α r ( R n ) f\in\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^{n}) f ∈ B ˙ ∞ s M p , α r ( R n ) 0 < p < 1 0<p<1 0 < p < 1 j ∈ Z j\in\mathbb{Z} j ∈ Z ψ j ∈ C ∞ ( R n ) \psi_j\in C^\infty(\mathbb{R}^{n}) ψ j ∈ C ∞ ( R n ) 0 ≤ ψ j ≤ 1 0\leq\psi_j\leq1 0 ≤ ψ j ≤ 1 ψ j = 1 \psi_j=1 ψ j = 1 ∣ ξ ∣ ≤ 2 j + 1 |\xi|\leq2^{j+1} ∣ ξ ∣ ≤ 2 j + 1 ψ j = 0 \psi_j=0 ψ j = 0 ∣ ξ ∣ ≥ 2 j + 2 |\xi|\geq2^{j+2} ∣ ξ ∣ ≥ 2 j + 2 φ j \varphi_j φ j (5) , as supp ( φ j ) ⊂ B 2 j + 1 ( 0 ) \operatorname{supp}(\varphi_j)\subset B_{2^{j+1}}(0) supp ( φ j ) ⊂ B 2 j + 1 ( 0 ) φ j = ψ j φ j \varphi_j=\psi_j\varphi_j φ j = ψ j φ j
φ j ( D ) f ( x ) = F − 1 ( ψ j φ j F f ) ( x ) = K j ∗ ( φ j ( D ) f ) ( x ) , (13) \varphi_j(D)f(x)=\mathcal{F}^{-1}(\psi_j\varphi_j\mathcal{F}f)(x)=K_j*(\varphi_j(D)f)(x), \tag*{(13)} φ j ( D ) f ( x ) = F − 1 ( ψ j φ j F f ) ( x ) = K j ∗ ( φ j ( D ) f ) ( x ) , (13) where K j = F − 1 ( ψ j ) K_j=\mathcal{F}^{-1}(\psi_j) K j = F − 1 ( ψ j )
∥ K j ∥ L ∞ ≲ 2 j n . (14) \|K_j\|_{L^\infty}\lesssim2^{jn}. \tag*{(14)} ∥ K j ∥ L ∞ ≲ 2 j n . (14) Now it follows from (13) that
∣ φ j ( D ) f ( x ) ∣ ≤ ∫ B 2 − j ( x ) ∣ K j ( x − y ) ∣ ∣ φ j ( D ) f ( y ) ∣ d y + ∑ i = j − 1 ∞ ∫ B 2 − i ( x ) ∖ B 2 − i − 1 ( x ) ∣ K j ( x − y ) ∣ ∣ φ j ( D ) f ( y ) ∣ d y = : I + I I . (15) \begin{aligned}
|\varphi_j(D)f(x)|
&\leq\int_{B_{2^{-j}}(x)}|K_j(x-y)||\varphi_j(D)f(y)|\,dy\\
&\quad+\sum_{i=j-1}^{\infty}\int_{B_{2^{-i}}(x)\setminus B_{2^{-i-1}}(x)}|K_j(x-y)||\varphi_j(D)f(y)|\,dy\\
&=:I+II. \tag*{(15)}
\end{aligned} ∣ φ j ( D ) f ( x ) ∣ ≤ ∫ B 2 − j ( x ) ∣ K j ( x − y ) ∣∣ φ j ( D ) f ( y ) ∣ d y + i = j − 1 ∑ ∞ ∫ B 2 − i ( x ) ∖ B 2 − i − 1 ( x ) ∣ K j ( x − y ) ∣∣ φ j ( D ) f ( y ) ∣ d y =: I + I I . (15) On the other hand, by (14) , we have
I ≲ 2 − j s p ∥ K j ∥ L ∞ ∥ φ j ( D ) f ∥ L ∞ 1 − p ∫ B 2 − j ( x ) ( 2 j s ∣ φ j ( D ) f ( y ) ∣ ) p d y I \lesssim2^{-jsp}\lVert K_j\rVert_{L^\infty}\lVert\varphi_j(D)f\rVert_{L^\infty}^{1-p}\int_{B_{2^{-j}}(x)}\left(2^{js}\lvert\varphi_j(D)f(y)\rvert\right)^p\,dy I ≲ 2 − j s p ∥ K j ∥ L ∞ ∥ φ j ( D ) f ∥ L ∞ 1 − p ∫ B 2 − j ( x ) ( 2 j s ∣ φ j ( D ) f ( y )∣ ) p d y ≲ 2 − j s p 2 j n ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p 2 − j ( n − α r ) \lesssim2^{-jsp}2^{jn}\lVert\varphi_j(D)f\rVert_{L^\infty}^{1-p}\lVert f\rVert_{\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^n)}^p2^{-j(n-\alpha r)} ≲ 2 − j s p 2 j n ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p 2 − j ( n − α r ) ≲ 2 − ( s p − α r ) j ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p . \lesssim2^{-(sp-\alpha r)j}\lVert\varphi_j(D)f\rVert_{L^\infty}^{1-p}\lVert f\rVert_{\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^n)}^p. ≲ 2 − ( s p − α r ) j ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p . Similarly, for each integer i ≥ j − 1 i\ge j-1 i ≥ j − 1
∫ B 2 − i ( x ) ∖ B 2 − i − 1 ( x ) ∣ K j ( x − y ) ∣ ∣ φ j ( D ) f ( y ) ∣ d y \int_{B_{2^{-i}}(x)\setminus B_{2^{-i-1}}(x)}\lvert K_j(x-y)\rvert\lvert\varphi_j(D)f(y)\rvert\,dy ∫ B 2 − i ( x ) ∖ B 2 − i − 1 ( x ) ∣ K j ( x − y )∣ ∣ φ j ( D ) f ( y )∣ d y ≲ 2 − j s p ∥ K j ∥ L ∞ ∥ φ j ( D ) f ∥ L ∞ 1 − p ∫ B 2 − i ( x ) ( 2 j s ∣ φ j ( D ) f ( y ) ∣ ) p d y \lesssim2^{-jsp}\lVert K_j\rVert_{L^\infty}\lVert\varphi_j(D)f\rVert_{L^\infty}^{1-p}\int_{B_{2^{-i}}(x)}\left(2^{js}\lvert\varphi_j(D)f(y)\rvert\right)^p\,dy ≲ 2 − j s p ∥ K j ∥ L ∞ ∥ φ j ( D ) f ∥ L ∞ 1 − p ∫ B 2 − i ( x ) ( 2 j s ∣ φ j ( D ) f ( y )∣ ) p d y ≲ 2 − j s p 2 j n ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p 2 − i ( n − α r ) . \lesssim2^{-jsp}2^{jn}\lVert\varphi_j(D)f\rVert_{L^\infty}^{1-p}\lVert f\rVert_{\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^n)}^p2^{-i(n-\alpha r)}. ≲ 2 − j s p 2 j n ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p 2 − i ( n − α r ) . Thus, it follows that
I I ≲ 2 − s j p 2 j n ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p ∑ i = j − 1 ∞ 2 − i ( n − α r ) II\lesssim2^{-sjp}2^{jn}\lVert\varphi_j(D)f\rVert_{L^\infty}^{1-p}\lVert f\rVert_{\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^n)}^p\sum_{i=j-1}^{\infty}2^{-i(n-\alpha r)} I I ≲ 2 − s j p 2 j n ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p i = j − 1 ∑ ∞ 2 − i ( n − α r ) ≲ 2 − ( s p − α r ) j ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p . \lesssim2^{-(sp-\alpha r)j}\lVert\varphi_j(D)f\rVert_{L^\infty}^{1-p}\lVert f\rVert_{\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^n)}^p. ≲ 2 − ( s p − α r ) j ∥ φ j ( D ) f ∥ L ∞ 1 − p ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) p . Assume for the moment that ∥ φ j ( D ) f ∥ L ∞ < + ∞ \lVert\varphi_j(D)f\rVert_{L^\infty}<+\infty ∥ φ j ( D ) f ∥ L ∞ < + ∞ (15) and the estimates for I I I I I II I I
∥ φ j ( D ) f ∥ L ∞ ≲ 2 − ( s − α r / p ) j ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) . \lVert\varphi_j(D)f\rVert_{L^\infty}\lesssim2^{-(s-\alpha r/p)j}\lVert f\rVert_{\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^n)}. ∥ φ j ( D ) f ∥ L ∞ ≲ 2 − ( s − α r / p ) j ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) . for all j ∈ Z j\in\mathbb{Z} j ∈ Z (12) as desired.
To remove the assumption ∥ φ j ( D ) f ∥ L ∞ < + ∞ \lVert\varphi_j(D)f\rVert_{L^\infty}<+\infty ∥ φ j ( D ) f ∥ L ∞ < + ∞ ρ ∈ C c ∞ ( B 1 ( 0 ) ) \rho\in C^\infty_c(B_1(0)) ρ ∈ C c ∞ ( B 1 ( 0 )) ∫ R n ρ d x = 1 \int_{\mathbb{R}^n}\rho\,dx=1 ∫ R n ρ d x = 1 h ( x ) = F − 1 ( ρ ) ( x ) h(x)=\mathcal{F}^{-1}(\rho)(x) h ( x ) = F − 1 ( ρ ) ( x ) h ( 0 ) = 1 h(0)=1 h ( 0 ) = 1 φ j ( D ) f \varphi_j(D)f φ j ( D ) f h ( t ⋅ ) φ j ( D ) f ∈ S ( R n ) ⊂ L ∞ ( R n ) h(t\cdot)\varphi_j(D)f\in\mathcal{S}(\mathbb{R}^n)\subset L^\infty(\mathbb{R}^n) h ( t ⋅ ) φ j ( D ) f ∈ S ( R n ) ⊂ L ∞ ( R n ) t > 0 t>0 t > 0
supp ( F ( h ( t ⋅ ) φ j ( D ) f ) ) ⊂ B t + 2 j + 1 ( 0 ) . \operatorname{supp}\big(\mathcal{F}(h(t\cdot)\varphi_j(D)f)\big)\subset B_{t+2^{j+1}}(0). supp ( F ( h ( t ⋅ ) φ j ( D ) f ) ) ⊂ B t + 2 j + 1 ( 0 ) . Thus, by modifying the argument above, we find, for any t > 0 t>0 t > 0 j ∈ Z j\in\mathbb{Z} j ∈ Z
∥ h ( t ⋅ ) φ j ( D ) f ∥ L ∞ ≲ ( t + 2 j ) n / p 2 − j n / p 2 − ( s − α r / p ) j ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) . \lVert h(t\cdot)\varphi_j(D)f\rVert_{L^\infty}\lesssim(t+2^j)^{n/p}2^{-jn/p}2^{-(s-\alpha r/p)j}\lVert f\rVert_{\dot{B}^{s}_{\infty}\mathcal{M}^{p,\alpha r}(\mathbb{R}^n)}. ∥ h ( t ⋅ ) φ j ( D ) f ∥ L ∞ ≲ ( t + 2 j ) n / p 2 − j n / p 2 − ( s − α r / p ) j ∥ f ∥ B ˙ ∞ s M p , α r ( R n ) . Then letting t → 0 t\to0 t → 0 (12) . □
We now recall the notion vanishing at infinity for tempered distributions in R n \mathbb{R}^n R n [4 Definition 4.3] ).
Definition 2.11. We say that a distribution f ∈ S ′ ( R n ) f\in\mathcal{S}'(\mathbb{R}^n) f ∈ S ′ ( R n )
lim λ → 0 ⟨ f , λ n φ ( λ ⋅ ) ⟩ = 0 , ∀ φ ∈ S ( R n ) . \lim_{\lambda\to0}\langle f,\lambda^n\varphi(\lambda\cdot)\rangle=0,\qquad\forall\ \varphi\in\mathcal{S}(\mathbb{R}^n). λ → 0 lim ⟨ f , λ n φ ( λ ⋅ )⟩ = 0 , ∀ φ ∈ S ( R n ) . The set of such distributions is denoted by S 0 ′ ( R n ) \mathcal{S}'_0(\mathbb{R}^n) S 0 ′ ( R n )
By definition, B ˙ ∞ , ∞ β ( R n ) \dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) B ˙ ∞ , ∞ β ( R n ) β ∈ R \beta\in\mathbb{R} β ∈ R S ′ / P ( R n ) \mathcal{S}'/\mathcal{P}(\mathbb{R}^{n}) S ′ / P ( R n ) β < 0 \beta< 0 β < 0 S ′ \mathcal{S}' S ′ B ˙ ∞ , ∞ β ( R n ) \dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) B ˙ ∞ , ∞ β ( R n ) β < 0 \beta< 0 β < 0 S ′ \mathcal{S}' S ′ [4 Proposition 4.6] ).
Proposition 2.12. Let β < 0 \beta< 0 β < 0 f ∈ B ˙ ∞ , ∞ β ( R n ) f \in\dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) f ∈ B ˙ ∞ , ∞ β ( R n ) ∑ j ∈ Z Δ ˙ j f \sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}f ∑ j ∈ Z Δ ˙ j f S ′ ( R n ) \mathcal{S}'(\mathbb{R}^{n}) S ′ ( R n )
Φ : B ˙ ∞ , ∞ β ( R n ) → S ′ ( R n ) given by Φ ( [ f ] ) : = ∑ j ∈ Z Δ ˙ j f \Phi: \dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) \to\mathcal{S}'(\mathbb{R}^{n}) \text{ given by } \Phi([f]) := \sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}f Φ : B ˙ ∞ , ∞ β ( R n ) → S ′ ( R n ) given by Φ ([ f ]) := j ∈ Z ∑ Δ ˙ j f is a realization of B ˙ ∞ , ∞ β ( R n ) \dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) B ˙ ∞ , ∞ β ( R n ) S ′ \mathcal{S}' S ′ Φ \Phi Φ Φ ( [ f ] ) ∈ [ f ] \Phi([f]) \in[f] Φ ([ f ]) ∈ [ f ] f f f [ f ] ∈ B ˙ ∞ , ∞ β ( R n ) [f] \in\dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) [ f ] ∈ B ˙ ∞ , ∞ β ( R n ) Φ ( B ˙ ∞ , ∞ β ( R n ) ) \Phi(\dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n})) Φ ( B ˙ ∞ , ∞ β ( R n )) g ∈ S ′ g \in\mathcal{S}' g ∈ S ′ [ g ] ∈ B ˙ ∞ , ∞ β ( R n ) [g] \in\dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) [ g ] ∈ B ˙ ∞ , ∞ β ( R n ) g g g
We remark that for g ∈ Φ ( B ˙ ∞ , ∞ β ( R n ) ) g \in\Phi(\dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n})) g ∈ Φ ( B ˙ ∞ , ∞ β ( R n )) g = ∑ j ∈ Z Δ ˙ j f g = \sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}f g = ∑ j ∈ Z Δ ˙ j f S ′ ( R n ) \mathcal{S}'(\mathbb{R}^{n}) S ′ ( R n ) f ∈ S ′ ( R n ) f \in\mathcal{S}'(\mathbb{R}^{n}) f ∈ S ′ ( R n ) ∑ j ∈ Z Δ ˙ j f = f + P \sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}f = f + P ∑ j ∈ Z Δ ˙ j f = f + P P P P g = ∑ j ∈ Z Δ ˙ j g g = \sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}g g = ∑ j ∈ Z Δ ˙ j g S ′ ( R n ) \mathcal{S}'(\mathbb{R}^{n}) S ′ ( R n )
Since F ˙ q s M p α , r ( R n ) ↪ B ˙ ∞ , ∞ s − α r p ( R n ) \dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n}) \hookrightarrow\dot{B}^{s-\frac{\alpha r}{p}}_{\infty,\infty}(\mathbb{R}^{n}) F ˙ q s M p α , r ( R n ) ↪ B ˙ ∞ , ∞ s − p α r ( R n ) F ˙ q s M p α , r ( R n ) \dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n}) F ˙ q s M p α , r ( R n ) S ′ ( R n ) \mathcal{S}'(\mathbb{R}^{n}) S ′ ( R n )
Corollary 2.13. Let ( s , α ) ∈ R × R + (s,\alpha) \in\mathbb{R} \times\mathbb{R}_{+} ( s , α ) ∈ R × R + ( p , q , r ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ] × ( 1 , ∞ ) (p,q,r) \in(0,\infty) \times(0,\infty] \times(1,\infty) ( p , q , r ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ] × ( 1 , ∞ ) α r < n \alpha r < n α r < n s < α r p s < \frac{\alpha r}{p} s < p α r
Φ : F ˙ q s M p α , r ( R n ) → S ′ ( R n ) given by Φ ( [ f ] ) : = ∑ j ∈ Z Δ ˙ j f \Phi: \dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n}) \to\mathcal{S}'(\mathbb{R}^{n}) \text{ given by } \Phi([f]) := \sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}f Φ : F ˙ q s M p α , r ( R n ) → S ′ ( R n ) given by Φ ([ f ]) := j ∈ Z ∑ Δ ˙ j f is a realization of F ˙ q s M p α , r ( R n ) \dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n}) F ˙ q s M p α , r ( R n ) S ′ \mathcal{S}' S ′ Φ ( F ˙ q s M p α , r ( R n ) ) \Phi(\dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n})) Φ ( F ˙ q s M p α , r ( R n )) g ∈ S ′ g \in\mathcal{S}' g ∈ S ′ [ g ] ∈ F ˙ q s M p α , r ( R n ) [g] \in\dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n}) [ g ] ∈ F ˙ q s M p α , r ( R n ) g g g
Remark 2.14. Let β < 0 \beta< 0 β < 0 E E E B ˙ ∞ , ∞ β ( R n ) \dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) B ˙ ∞ , ∞ β ( R n ) E E E Φ ( E ) \Phi(E) Φ ( E ) E E E S ′ ( R n ) \mathcal{S}'(\mathbb{R}^{n}) S ′ ( R n ) B ˙ ∞ , ∞ β ( R n ) \dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n}) B ˙ ∞ , ∞ β ( R n ) Φ ( B ˙ ∞ , ∞ β ( R n ) ) \Phi(\dot{B}^{\beta}_{\infty,\infty}(\mathbb{R}^{n})) Φ ( B ˙ ∞ , ∞ β ( R n )) F ˙ q s M p α , r ( R n ) \dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n}) F ˙ q s M p α , r ( R n ) Φ ( F ˙ q s M p α , r ( R n ) ) \Phi(\dot{F}^{s}_{q}M^{\alpha,r}_{p}(\mathbb{R}^{n})) Φ ( F ˙ q s M p α , r ( R n )) s < α r p s < \frac{\alpha r}{p} s < p α r
We will need the following characterizations of weighted homogeneous Triebel-Lizorkin spaces with negative order when the weights involved are in the A ∞ A_{\infty} A ∞
Theorem 2.15. Let s < 0 s < 0 s < 0 ( p , q ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ] (p,q) \in(0,\infty) \times(0,\infty] ( p , q ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ] w ∈ A ∞ w \in A_{\infty} w ∈ A ∞ ν \nu ν S ( R n ) \mathcal{S}(\mathbb{R}^{n}) S ( R n )
∀ ξ ≠ 0 there exists t > 0 such that F ( ν ) ( t ξ ) ≠ 0. (16) \forall\, \xi\ne0 \quad\text{there exists } t > 0 \text{ such that } \mathcal{F}(\nu)(t\xi) \ne0. \tag*{(16)} ∀ ξ = 0 there exists t > 0 such that F ( ν ) ( t ξ ) = 0. (16) :::`
Then we have
∥ f ∥ F ˙ q s L p ( w ) ≤ C ∥ ( ∫ 0 ∞ ∣ t − s ⟨ f , ν t ( x − ⋅ ) ⟩ ∣ q d t t ) 1 q ∥ L p ( w ) , (17) \|f\|_{\dot{F}^{s}_{q}L^{p}(w)} \leq C \left\|\left(\int_{0}^{\infty} |t^{-s}\langle f,\nu_t(x-\cdot)\rangle|^{q}\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{L^{p}(w)}, \tag*{(17)} ∥ f ∥ F ˙ q s L p ( w ) ≤ C ( ∫ 0 ∞ ∣ t − s ⟨ f , ν t ( x − ⋅ )⟩ ∣ q t d t ) q 1 L p ( w ) , (17) where ν t ( ⋅ ) = t − n ν ( ⋅ / t ) \nu_t(\cdot)=t^{-n}\nu(\cdot/t) ν t ( ⋅ ) = t − n ν ( ⋅ / t ) P P P
∥ ( ∫ 0 ∞ ∣ t − s ⟨ f − P , ν t ( x − ⋅ ) ⟩ ∣ q d t t ) 1 q ∥ L p ( w ) ≤ C ∥ f ∥ F ˙ q s L p ( w ) . (18) \left\|\left(\int_{0}^{\infty} |t^{-s}\langle f-P,\nu_t(x-\cdot)\rangle|^{q}\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{L^{p}(w)} \leq C\|f\|_{\dot{F}^{s}_{q}L^{p}(w)}. \tag*{(18)} ( ∫ 0 ∞ ∣ t − s ⟨ f − P , ν t ( x − ⋅ )⟩ ∣ q t d t ) q 1 L p ( w ) ≤ C ∥ f ∥ F ˙ q s L p ( w ) . (18) With an appropriate modification, inequalities (17) and (18) also hold for q = ∞ q=\infty q = ∞
Inequalities (17) and (18) were obtained in [10 Theorem 2] and [9 Theorem 3.1] , respectively (see also [7 ] and [8 ] ). We notice from the proof of [9 Theorem 3.1] that one can take P = 0 P=0 P = 0 f f f
f = ∑ j ∈ Z Δ ˙ j f in S ′ ( R n ) . (19) f=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}f\quad\text{in }\mathcal{S}'(\mathbb{R}^{n}). \tag*{(19)} f = j ∈ Z ∑ Δ ˙ j f in S ′ ( R n ) . (19) We now use Theorem Theorem 2.15 and Proposition Proposition 2.8 to obtain useful characterizations of F ˙ q s , α , r M p ( R n ) \dot{F}^{s,\alpha,r}_{q}M_{p}(\mathbb{R}^{n}) F ˙ q s , α , r M p ( R n ) s < 0 s<0 s < 0
Theorem 2.16. Let s < 0 s<0 s < 0 p ∈ ( 0 , ∞ ) p\in(0,\infty) p ∈ ( 0 , ∞ ) q ∈ ( 0 , ∞ ] q\in(0,\infty] q ∈ ( 0 , ∞ ] r ∈ ( 1 , ∞ ) r\in(1,\infty) r ∈ ( 1 , ∞ ) α > 0 \alpha>0 α > 0 α r < n \alpha r<n α r < n ν ∈ C c ∞ ( R n ) \nu\in C^{\infty}_{c}(\mathbb{R}^{n}) ν ∈ C c ∞ ( R n ) χ B 1 / 2 ( 0 ) ≤ ν ≤ χ B 1 ( 0 ) \chi_{B_{1/2}(0)}\leq\nu\leq\chi_{B_{1}(0)} χ B 1/2 ( 0 ) ≤ ν ≤ χ B 1 ( 0 ) ν t ( ⋅ ) = t − n ν ( ⋅ / t ) \nu_t(\cdot)=t^{-n}\nu(\cdot/t) ν t ( ⋅ ) = t − n ν ( ⋅ / t ) t > 0 t>0 t > 0
∥ f ∥ F ˙ q s , α , r M p ( R n ) ≲ ∥ ( ∫ 0 ∞ ∣ t − s ⟨ f , ν t ( x − ⋅ ) ⟩ ∣ q d t t ) 1 q ∥ M p α , r ( R n ) , (20) \|f\|_{\dot{F}^{s,\alpha,r}_{q}M_{p}(\mathbb{R}^{n})}\lesssim\left\|\left(\int_{0}^{\infty}|t^{-s}\langle f,\nu_t(x-\cdot)\rangle|^{q}\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{M^{\alpha,r}_{p}(\mathbb{R}^{n})}, \tag*{(20)} ∥ f ∥ F ˙ q s , α , r M p ( R n ) ≲ ( ∫ 0 ∞ ∣ t − s ⟨ f , ν t ( x − ⋅ )⟩ ∣ q t d t ) q 1 M p α , r ( R n ) , (20) ∥ f ∥ F ˙ q s , α , r M p ( R n ) ≲ ∥ ( ∫ 0 ∞ ∣ t − s 2 t Δ f ( x ) ∣ q d t t ) 1 q ∥ M p α , r ( R n ) , (21) \|f\|_{\dot{F}^{s,\alpha,r}_{q}M_{p}(\mathbb{R}^{n})}\lesssim\left\|\left(\int_{0}^{\infty}|t^{-\frac{s}{2}}t\Delta f(x)|^{q}\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{M^{\alpha,r}_{p}(\mathbb{R}^{n})}, \tag*{(21)} ∥ f ∥ F ˙ q s , α , r M p ( R n ) ≲ ( ∫ 0 ∞ ∣ t − 2 s t Δ f ( x ) ∣ q t d t ) q 1 M p α , r ( R n ) , (21) and there exists a polynomial P P P
∥ ( ∫ 0 ∞ ∣ t − s ⟨ f − P , ν t ( x − ⋅ ) ⟩ ∣ q d t t ) 1 q ∥ M p α , r ( R n ) ≲ ∥ f ∥ F ˙ q s , α , r M p ( R n ) , (22) \left\|\left(\int_{0}^{\infty}|t^{-s}\langle f-P,\nu_t(x-\cdot)\rangle|^{q}\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{M^{\alpha,r}_{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{F}^{s,\alpha,r}_{q}M_{p}(\mathbb{R}^{n})}, \tag*{(22)} ( ∫ 0 ∞ ∣ t − s ⟨ f − P , ν t ( x − ⋅ )⟩ ∣ q t d t ) q 1 M p α , r ( R n ) ≲ ∥ f ∥ F ˙ q s , α , r M p ( R n ) , (22) ∥ ( ∫ 0 ∞ ∣ t − s 2 t Δ ( f − P ) ∣ q d t t ) 1 q ∥ M p α , r ( R n ) ≲ ∥ f ∥ F ˙ q s , α , r M p ( R n ) . (23) \left\|\left(\int_{0}^{\infty}|t^{-\frac{s}{2}}t\Delta(f-P)|^{q}\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{M^{\alpha,r}_{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{F}^{s,\alpha,r}_{q}M_{p}(\mathbb{R}^{n})}. \tag*{(23)} ( ∫ 0 ∞ ∣ t − 2 s t Δ ( f − P ) ∣ q t d t ) q 1 M p α , r ( R n ) ≲ ∥ f ∥ F ˙ q s , α , r M p ( R n ) . (23) With an appropriate modification, inequalities (20) –(23) also hold for q = ∞ q=\infty q = ∞ P = 0 P=0 P = 0 (22) and (23) when f f f (19) .
Proof. Inequalities (20) and (22) follow from Proposition 2.8 and Theorem 2.15 with ν ∈ C c ∞ ( R n ) \nu\in C^\infty_c(\mathbb{R}^n) ν ∈ C c ∞ ( R n ) χ B 1 / 2 ( 0 ) ≤ ν ≤ χ B 1 ( 0 ) \chi_{B_{1/2}(0)} \leq\nu\leq\chi_{B_1(0)} χ B 1/2 ( 0 ) ≤ ν ≤ χ B 1 ( 0 ) (16) holds in this case. Similarly, inequalities (21) and (23) follow from Proposition 2.8 and Theorem 2.15, with ν = ( 4 π ) − n / 2 e − ∣ ⋅ ∣ 2 / 4 \nu=(4\pi)^{-n/2}e^{-|\cdot|^2/4} ν = ( 4 π ) − n /2 e − ∣ ⋅ ∣ 2 /4 □ \square □
Semilinear parabolic equations with nonnegative measures as initial data
We now recall the following fundamental result on the existence of nonnegative global in time solutions to the Cauchy problem
{ ∂ t u − Δ u = ∣ u ∣ b − 1 u in R n × ( 0 , ∞ ) , u ( ⋅ , 0 ) = u 0 in R n , (24) \begin{cases}
\partial_t u-\Delta u=|u|^{b-1}u & \text{in }\mathbb{R}^n\times(0,\infty),\\
u(\cdot,0)=u_0 & \text{in }\mathbb{R}^n, \tag*{(24)}
\end{cases} { ∂ t u − Δ u = ∣ u ∣ b − 1 u u ( ⋅ , 0 ) = u 0 in R n × ( 0 , ∞ ) , in R n , (24) with nonnegative measure initial data that was obtained by Baras and Pierre in [3 Theorem 2.1] (see also [20 Theorem 2.12] ).
Theorem 3.1. Let b > 1 b>1 b > 1 u 0 u_0 u 0 R n \mathbb{R}^n R n (24) has a nonnegative solution u ∈ L l o c b ( R n × ( 0 , ∞ ) ) u\in L^b_{\mathrm{loc}}(\mathbb{R}^n\times(0,\infty)) u ∈ L loc b ( R n × ( 0 , ∞ ))
∫ 0 ∞ ∫ R n e t Δ u 0 ( x ) h ( x , t ) d x d t ≤ b − 1 b b ′ ∫ 0 ∞ ∫ R n h ( x , t ) b ′ [ H ~ ∗ h ( x , t ) ] b ′ − 1 d x d t \int_0^\infty\int_{\mathbb{R}^n}e^{t\Delta}u_0(x)h(x,t)\,dx\,dt\leq\frac{b-1}{b^{b'}}\int_0^\infty\int_{\mathbb{R}^n}\frac{h(x,t)^{b'}}{[\widetilde{H}*h(x,t)]^{b'-1}}\,dx\,dt ∫ 0 ∞ ∫ R n e t Δ u 0 ( x ) h ( x , t ) d x d t ≤ b b ′ b − 1 ∫ 0 ∞ ∫ R n [ H ∗ h ( x , t ) ] b ′ − 1 h ( x , t ) b ′ d x d t holds for all nonnegative h ∈ L c ∞ ( R n × ( 0 , ∞ ) ) h\in L^\infty_c(\mathbb{R}^n\times(0,\infty)) h ∈ L c ∞ ( R n × ( 0 , ∞ )) R n × ( 0 , ∞ ) \mathbb{R}^n\times(0,\infty) R n × ( 0 , ∞ )
H ~ ( x , t ) = H ( x , − t ) = χ ( − ∞ , 0 ) ( t ) ( − 4 π t ) n / 2 e ∣ x ∣ 2 4 t , ( x , t ) ∈ R n + 1 , \widetilde{H}(x,t)=H(x,-t)=\frac{\chi_{(-\infty,0)}(t)}{(-4\pi t)^{n/2}}e^{\frac{|x|^2}{4t}},\qquad(x,t)\in\mathbb{R}^{n+1}, H ( x , t ) = H ( x , − t ) = ( − 4 π t ) n /2 χ ( − ∞ , 0 ) ( t ) e 4 t ∣ x ∣ 2 , ( x , t ) ∈ R n + 1 , is the adjoint of the heat kernel H ( x , t ) H(x,t) H ( x , t )
We will need the following lemma but only in the case p > 1 p>1 p > 1
Lemma 3.2. Let α , s ∈ ( 0 , n ) \alpha,s\in(0,n) α , s ∈ ( 0 , n ) p > 0 p>0 p > 0 1 < r < n α 1<r<\frac{n}{\alpha} 1 < r < α n μ \mu μ R n \mathbb{R}^n R n
c 1 ∥ I s [ μ ] ∥ M p α , r ( R n ) ≤ ∥ μ ∥ F ˙ q − s M p α , r ( R n ) ≤ c 2 ∥ I s [ μ ] ∥ M p α , r ( R n ) (25) c_1\lVert I_s[\mu]\rVert_{\mathcal{M}^{\alpha,r}_p(\mathbb{R}^n)}\leq\lVert\mu\rVert_{\dot{F}^{-s}_{q}\mathcal{M}^{\alpha,r}_p(\mathbb{R}^n)}\leq c_2\lVert I_s[\mu]\rVert_{\mathcal{M}^{\alpha,r}_p(\mathbb{R}^n)} \tag*{(25)} c 1 ∥ I s [ μ ] ∥ M p α , r ( R n ) ≤ ∥ μ ∥ F ˙ q − s M p α , r ( R n ) ≤ c 2 ∥ I s [ μ ] ∥ M p α , r ( R n ) (25) for all 0 < q ≤ ∞ 0<q\leq\infty 0 < q ≤ ∞ μ \mu μ
μ = ∑ j ∈ Z Δ j μ in S ′ ( R n ) . \mu=\sum_{j\in\mathbb{Z}}\Delta_j\mu\quad\text{in }\mathcal{S}'(\mathbb{R}^n). μ = j ∈ Z ∑ Δ j μ in S ′ ( R n ) . Proof. We first recall the weighted bound
∥ ( ∫ 0 ∞ ( μ ( B t ( ⋅ ) ) t n − s ) ρ d t t ) 1 / ρ ∥ L p ( w ) ≲ ∥ sup t > 0 μ ( B t ( ⋅ ) ) t n − s ∥ L p ( w ) (26) \left\lVert\left(\int_0^\infty\left(\frac{\mu(B_t(\cdot))}{t^{n-s}}\right)^\rho\frac{dt}{t}\right)^{1/\rho}\right\rVert_{L^p(w)}
\lesssim\left\lVert\sup_{t>0}\frac{\mu(B_t(\cdot))}{t^{n-s}}\right\rVert_{L^p(w)} \tag*{(26)} ( ∫ 0 ∞ ( t n − s μ ( B t ( ⋅ )) ) ρ t d t ) 1/ ρ L p ( w ) ≲ t > 0 sup t n − s μ ( B t ( ⋅ )) L p ( w ) (26) which holds for all 0 < p , ρ < ∞ 0<p,\rho<\infty 0 < p , ρ < ∞ w ∈ A ∞ w\in A_\infty w ∈ A ∞ ρ ≥ 1 \rho\geq1 ρ ≥ 1 [32 ] and the case 0 < ρ < 1 0<\rho<1 0 < ρ < 1 [19 ] and [18 ] .
Let ν ∈ C ∞ ( R n ) \nu\in C^\infty(\mathbb{R}^n) ν ∈ C ∞ ( R n ) χ B 1 / 2 ( 0 ) ≤ ν ≤ χ B 1 ( 0 ) \chi_{B_{1/2}(0)} \le\nu\le\chi_{B_1(0)} χ B 1/2 ( 0 ) ≤ ν ≤ χ B 1 ( 0 ) ν t ( ⋅ ) = t − n ν ( ⋅ / t ) \nu_t(\cdot)=t^{-n}\nu(\cdot/t) ν t ( ⋅ ) = t − n ν ( ⋅ / t ) (25) , it is enough to consider the case q = ∞ q=\infty q = ∞ f = μ ≥ 0 f=\mu\ge0 f = μ ≥ 0 ν \nu ν
∥ μ ∥ F ˙ ∞ − s L p ( w ) ≥ C 1 ∥ sup t > 0 μ ( B t ( ⋅ ) ) t n − s ∥ L p ( w ) ≥ C 2 ∥ I s [ μ ] ∥ L p ( w ) (27) \|\mu\|_{\dot{F}^{-s}_{\infty}L^p(w)} \ge C_1 \left\|\sup_{t>0}\frac{\mu(B_t(\cdot))}{t^{n-s}}\right\|_{L^p(w)} \ge C_2\|\mathcal{I}_s[\mu]\|_{L^p(w)} \tag*{(27)} ∥ μ ∥ F ˙ ∞ − s L p ( w ) ≥ C 1 t > 0 sup t n − s μ ( B t ( ⋅ )) L p ( w ) ≥ C 2 ∥ I s [ μ ] ∥ L p ( w ) (27) where w ∈ A ∞ w\in A_\infty w ∈ A ∞ (26) with ρ = 1 \rho=1 ρ = 1 A 1 ⊂ A ∞ A_1\subset A_\infty A 1 ⊂ A ∞ (27) we obtain the first bound in (25) .
Conversely, in order to prove the second inequality in (25) , we may assume that q ≠ ∞ q\ne\infty q = ∞ w ∈ A ∞ w\in A_\infty w ∈ A ∞ f = μ ≥ 0 f=\mu\ge0 f = μ ≥ 0
∥ μ ∥ F ˙ q − s L p ( w ) ≲ ∥ ( ∫ 0 ∞ ( μ ( B t ( ⋅ ) ) t n − s ) q d t t ) 1 q ∥ L p ( w ) \|\mu\|_{\dot{F}^{-s}_{q}L^p(w)} \lesssim\left\|\left(\int_0^\infty\left(\frac{\mu(B_t(\cdot))}{t^{n-s}}\right)^q\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{L^p(w)} ∥ μ ∥ F ˙ q − s L p ( w ) ≲ ( ∫ 0 ∞ ( t n − s μ ( B t ( ⋅ )) ) q t d t ) q 1 L p ( w ) ≲ ∥ sup t > 0 μ ( B t ( ⋅ ) ) t n − s ∥ L p ( w ) \lesssim\left\|\sup_{t>0}\frac{\mu(B_t(\cdot))}{t^{n-s}}\right\|_{L^p(w)} ≲ t > 0 sup t n − s μ ( B t ( ⋅ )) L p ( w ) ≲ ∥ I s [ μ ] ∥ L p ( w ) , (28) \lesssim\|\mathcal{I}_s[\mu]\|_{L^p(w)}, \tag*{(28)} ≲ ∥ I s [ μ ] ∥ L p ( w ) , (28) where we used (26) with ρ = q \rho=q ρ = q (28) and Proposition 2.8, we obtain the second bound in (25) as desired. □
Our main result in this section is the following.
Theorem 3.3. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 u 0 u_0 u 0 R n \mathbb{R}^n R n
(i) The equation
{ ∂ t u − Δ u = ∣ u ∣ b − 1 u in R n × ( 0 , ∞ ) , u ( ⋅ , 0 ) = κ u 0 in R n , (29) \begin{cases}
\partial_tu-\Delta u=|u|^{b-1}u & \text{in }\mathbb{R}^n\times(0,\infty),\\
u(\cdot,0)=\kappa u_0 & \text{in }\mathbb{R}^n, \tag*{(29)}
\end{cases} { ∂ t u − Δ u = ∣ u ∣ b − 1 u u ( ⋅ , 0 ) = κ u 0 in R n × ( 0 , ∞ ) , in R n , (29) has a nonnegative solution u ∈ L l o c b ( R n × ( 0 , ∞ ) ) u\in L^b_{\mathrm{loc}}(\mathbb{R}^n\times(0,\infty)) u ∈ L loc b ( R n × ( 0 , ∞ )) κ > 0 \kappa>0 κ > 0
(ii) There exists a constant c 1 > 0 c_1>0 c 1 > 0
u 0 ( K ) ≤ c 1 Cap 2 b , b ′ ( K ) u_0(K)\le c_1\operatorname{Cap}_{\frac{2}{b},b'}(K) u 0 ( K ) ≤ c 1 Cap b 2 , b ′ ( K ) holds for all compact sets K ⊂ R n K\subset\mathbb{R}^n K ⊂ R n
(iii) There exists a constant c 2 > 0 c_2>0 c 2 > 0
( u 0 ⊗ δ 0 ) ( E ) ≤ c 2 cap 2 , 1 , b ′ ( E ) (u_0\otimes\delta_0)(E)\le c_2\operatorname{cap}_{2,1,b'}(E) ( u 0 ⊗ δ 0 ) ( E ) ≤ c 2 cap 2 , 1 , b ′ ( E ) holds for all compact sets E ⊂ R n + 1 E\subset\mathbb{R}^{n+1} E ⊂ R n + 1 δ 0 \delta_0 δ 0 0 0 0 R \mathbb{R} R
(iv) There exists a constant c 3 > 0 c_3>0 c 3 > 0
∫ E I 2 ( u 0 ⊗ δ 0 ) b d x d t ≤ c 3 cap 2 , 1 , b ′ ( E ) \int_E \mathcal{I}_2(u_0\otimes\delta_0)^b\,dx\,dt\le c_3\operatorname{cap}_{2,1,b'}(E) ∫ E I 2 ( u 0 ⊗ δ 0 ) b d x d t ≤ c 3 cap 2 , 1 , b ′ ( E ) holds for all compact sets E ⊂ R n + 1 E \subset\mathbb{R}^{n+1} E ⊂ R n + 1
(v) There exists a constant c 4 > 0 c_4>0 c 4 > 0
∫ E [ e t Δ u 0 ( x ) ] b d x d t ≤ c 4 cap 2 , 1 , b ′ ( E ) \int_E \left[e^{t\Delta}u_0(x)\right]^b\,dxdt \le c_4\operatorname{cap}_{2,1,b'}(E) ∫ E [ e t Δ u 0 ( x ) ] b d x d t ≤ c 4 cap 2 , 1 , b ′ ( E ) holds for all compact sets E ⊂ R n + 1 E \subset\mathbb{R}^{n+1} E ⊂ R n + 1
(vi) There exists a constant c 5 > 0 c_5>0 c 5 > 0
∫ K ( I 2 b [ u 0 ] ) b d x ≤ c 5 Cap 2 b , b ′ ( K ) (30) \int_K \left(\mathrm{I}_{\frac{2}{b}}[u_0]\right)^b\,dx \le c_5\operatorname{Cap}_{\frac{2}{b},b'}(K) \tag*{(30)} ∫ K ( I b 2 [ u 0 ] ) b d x ≤ c 5 Cap b 2 , b ′ ( K ) (30) holds for all compact sets K ⊂ R n K \subset\mathbb{R}^n K ⊂ R n
(vii) u 0 u_0 u 0
u 0 = ∑ j ∈ Z Δ ˙ j u 0 in S ′ ( R n ) (31) u_0=\sum_{j\in\mathbb{Z}}\dot{\Delta}_j u_0 \quad\text{in } \mathcal{S}'(\mathbb{R}^n) \tag*{(31)} u 0 = j ∈ Z ∑ Δ ˙ j u 0 in S ′ ( R n ) (31) and u 0 ∈ F ˙ q − 2 b M b 2 , b ′ ( R n ) u_0\in\dot{F}^{-\frac{2}{b}}_{q}M^{2,b'}_b(\mathbb{R}^n) u 0 ∈ F ˙ q − b 2 M b 2 , b ′ ( R n ) 0 < q ≤ ∞ 0<q\le\infty 0 < q ≤ ∞
Proof. The equivalence (ii) ⇔ \Leftrightarrow ⇔ Theorem 2.2 and the fact that
cap H , b ′ ( K × { 0 } ) ≃ Cap 2 b , b ′ ( K ) (32) \operatorname{cap}_{H,b'}(K\times\{0\})\simeq\operatorname{Cap}_{\frac{2}{b},b'}(K) \tag*{(32)} cap H , b ′ ( K × { 0 }) ≃ Cap b 2 , b ′ ( K ) (32) for all compact sets K ⊂ R n K\subset\mathbb{R}^n K ⊂ R n [34 Corollary 2.20] ). That (iii) ⇔ \Leftrightarrow ⇔ Theorem 2.2 and [34 Theorem 2.36] . The implication (iv) ⇒ \Rightarrow ⇒ Theorem 2.2 the implication (v) ⇒ \Rightarrow ⇒ [34 Corollary 2.42] . Moreover, the equivalence (ii) ⇔ \Leftrightarrow ⇔ [27 Theorem 2.1] . Thus, it remains to show (i) ⇒ \Rightarrow ⇒ ⇔ \Leftrightarrow ⇔
• Proof of (i) ⇒ \Rightarrow ⇒ Suppose that (3.6), with some κ > 0 \kappa>0 κ > 0 u ∈ L l o c q ( R n × ( 0 , ∞ ) ) u\in L^q_{\mathrm{loc}}(\mathbb{R}^n\times(0,\infty)) u ∈ L loc q ( R n × ( 0 , ∞ )) Theorem 3.1 we have
∫ 0 ∞ ∫ R n e t Δ u 0 ( x ) h ( x , t ) d x d t ≤ b − 1 κ b b ′ ∫ 0 ∞ ∫ R n h ( x , t ) b ′ [ H ~ ∗ h ( x , t ) ] b ′ − 1 d x d t (33) \int_0^\infty\int_{\mathbb{R}^n}e^{t\Delta}u_0(x)h(x,t)\,dxdt\le\frac{b-1}{\kappa b^{b'}}\int_0^\infty\int_{\mathbb{R}^n}\frac{h(x,t)^{b'}}{[\widetilde{H}*h(x,t)]^{b'-1}}\,dxdt \tag*{(33)} ∫ 0 ∞ ∫ R n e t Δ u 0 ( x ) h ( x , t ) d x d t ≤ κ b b ′ b − 1 ∫ 0 ∞ ∫ R n [ H ∗ h ( x , t ) ] b ′ − 1 h ( x , t ) b ′ d x d t (33) for all nonnegative h ∈ L ∞ ( R n × ( 0 , ∞ ) ) h\in L^\infty(\mathbb{R}^n\times(0,\infty)) h ∈ L ∞ ( R n × ( 0 , ∞ )) (33) also holds for all h ∈ L c b ′ ( R n × [ 0 , ∞ ) ) h\in L^{b'}_c(\mathbb{R}^n\times[0,\infty)) h ∈ L c b ′ ( R n × [ 0 , ∞ )) h ≥ 0 h\ge0 h ≥ 0 h ∈ L c b ′ ( R n × [ 0 , ∞ ) ) h\in L^{b'}_c(\mathbb{R}^n\times[0,\infty)) h ∈ L c b ′ ( R n × [ 0 , ∞ )) h ≥ 0 h\ge0 h ≥ 0
v ( x , t ) = H ~ ∗ h ( x , t ) . v(x,t)=\widetilde{H}*h(x,t). v ( x , t ) = H ∗ h ( x , t ) . As − ∂ t v − Δ v = h -\partial_t v-\Delta v=h − ∂ t v − Δ v = h Theorem 3.1 and Fubini’s theorem, we have
∫ R n v ( ⋅ , 0 ) d u 0 = ∫ 0 ∞ ∫ R n e t Δ u 0 ( x ) h ( x , t ) d x d t ≤ b − 1 κ b b ′ ∫ 0 ∞ ∫ R n ( − ∂ t v − Δ v ) b ′ v b ′ − 1 d x d t . (34) \int_{\mathbb{R}^n}v(\cdot,0)\,du_0=\int_0^\infty\int_{\mathbb{R}^n}e^{t\Delta}u_0(x)h(x,t)\,dxdt
\le\frac{b-1}{\kappa b^{b'}}\int_0^\infty\int_{\mathbb{R}^n}\frac{(-\partial_t v-\Delta v)^{b'}}{v^{b'-1}}\,dxdt. \tag*{(34)} ∫ R n v ( ⋅ , 0 ) d u 0 = ∫ 0 ∞ ∫ R n e t Δ u 0 ( x ) h ( x , t ) d x d t ≤ κ b b ′ b − 1 ∫ 0 ∞ ∫ R n v b ′ − 1 ( − ∂ t v − Δ v ) b ′ d x d t . (34) We now claim that we can deduce from (34) the following bound
∫ R n Φ ( ⋅ , 0 ) 2 b ′ d u 0 ≤ c ( n , b , κ ) ∫ 0 ∞ ∫ R n ∣ ∂ t Φ ∣ b ′ + ∣ D 2 Φ ∣ b ′ d x d t (35) \int_{\mathbb{R}^n}\Phi(\cdot,0)^{2b'}\,du_0\leq c(n,b,\kappa)\int_0^\infty\int_{\mathbb{R}^n}\left|\partial_t\Phi\right|^{b'}+\left|D^2\Phi\right|^{b'}\,dxdt \tag*{(35)} ∫ R n Φ ( ⋅ , 0 ) 2 b ′ d u 0 ≤ c ( n , b , κ ) ∫ 0 ∞ ∫ R n ∣ ∂ t Φ ∣ b ′ + D 2 Φ b ′ d x d t (35) for all Φ ∈ C c ∞ ( R n + 1 ) \Phi\in C^\infty_c(\mathbb{R}^{n+1}) Φ ∈ C c ∞ ( R n + 1 ) 0 ≤ Φ ≤ 1 0\leq\Phi\leq1 0 ≤ Φ ≤ 1
To this end, let φ k = φ ( x / k , t / k ) \varphi_k=\varphi(x/k,t/k) φ k = φ ( x / k , t / k ) k = 1 , 2 , … k=1,2,\ldots k = 1 , 2 , … φ \varphi φ C c ∞ ( B 2 ( 0 ) ) C^\infty_c(B_2(0)) C c ∞ ( B 2 ( 0 )) 0 ≤ φ ≤ 1 0\leq\varphi\leq1 0 ≤ φ ≤ 1 φ = 1 \varphi=1 φ = 1 B 1 ( 0 ) B_1(0) B 1 ( 0 )
∣ ∂ t φ k ∣ + ∣ ∇ φ k ∣ + k ∣ Δ φ k ∣ ≤ C k . \left|\partial_t\varphi_k\right|+\left|\nabla\varphi_k\right|+k\left|\Delta\varphi_k\right|\leq\frac{C}{k}. ∣ ∂ t φ k ∣ + ∣ ∇ φ k ∣ + k ∣ Δ φ k ∣ ≤ k C . The applying (34) with v = H ~ ∗ h ≥ Φ k 2 q ′ Φ 2 q ′ v=\widetilde{H}*h\geq\Phi_k^{2q'}\Phi^{2q'} v = H ∗ h ≥ Φ k 2 q ′ Φ 2 q ′ R n × [ 0 , ∞ ) \mathbb{R}^n\times[0,\infty) R n × [ 0 , ∞ ) h = ∣ ( − ∂ t − Δ ) ( Φ 2 q ′ φ k 2 q ′ ) ∣ χ [ 0 , ∞ ) ( t ) h=\left|(-\partial_t-\Delta)(\Phi^{2q'}\varphi_k^{2q'})\right|\chi_{[0,\infty)}(t) h = ( − ∂ t − Δ ) ( Φ 2 q ′ φ k 2 q ′ ) χ [ 0 , ∞ ) ( t )
( − ∂ t − Δ ) v = h ≤ ∣ ( ∂ t + Δ ) ( Φ 2 q ′ ) φ k 2 q ′ ∣ + ∣ ( ∂ t + Δ ) ( φ k 2 q ′ ) Φ 2 q ′ + 2 ∣ ∇ ( Φ 2 q ′ ) ⋅ ∇ ( φ k 2 q ′ ) ∣ , (-\partial_t-\Delta)v=h\leq|(\partial_t+\Delta)(\Phi^{2q'})\varphi_k^{2q'}|
+|(\partial_t+\Delta)(\varphi_k^{2q'})\Phi^{2q'}+2|\nabla(\Phi^{2q'})\cdot\nabla(\varphi_k^{2q'})|, ( − ∂ t − Δ ) v = h ≤ ∣ ( ∂ t + Δ ) ( Φ 2 q ′ ) φ k 2 q ′ ∣ + ∣ ( ∂ t + Δ ) ( φ k 2 q ′ ) Φ 2 q ′ + 2∣∇ ( Φ 2 q ′ ) ⋅ ∇ ( φ k 2 q ′ ) ∣ , we find
∫ R n [ ( Φ φ k ) ( ⋅ , 0 ) ] 2 b ′ d u 0 ≲ ∫ 0 ∞ ∫ R n ∣ ( ∂ t + Δ ) ( Φ 2 b ′ ) ∣ b ′ φ k 2 b ′ Φ 2 b ′ ( 1 − b ′ ) d x d t \int_{\mathbb{R}^n}\left[(\Phi\varphi_k)(\cdot,0)\right]^{2b'}\,du_0
\lesssim\int_0^\infty\int_{\mathbb{R}^n}\left|(\partial_t+\Delta)(\Phi^{2b'})\right|^{b'}\varphi_k^{2b'}\Phi^{2b'(1-b')}\,dxdt ∫ R n [ ( Φ φ k ) ( ⋅ , 0 ) ] 2 b ′ d u 0 ≲ ∫ 0 ∞ ∫ R n ( ∂ t + Δ ) ( Φ 2 b ′ ) b ′ φ k 2 b ′ Φ 2 b ′ ( 1 − b ′ ) d x d t + ∫ 0 ∞ ∫ R n ∣ ( ∂ t + Δ ) ( φ k 2 b ′ ) ∣ b ′ φ k 2 b ′ ( 1 − b ′ ) Φ 2 b ′ d x d t \qquad+\int_0^\infty\int_{\mathbb{R}^n}\left|(\partial_t+\Delta)(\varphi_k^{2b'})\right|^{b'}\varphi_k^{2b'(1-b')}\Phi^{2b'}\,dxdt + ∫ 0 ∞ ∫ R n ( ∂ t + Δ ) ( φ k 2 b ′ ) b ′ φ k 2 b ′ ( 1 − b ′ ) Φ 2 b ′ d x d t + ∫ 0 ∞ ∫ R n ∣ ∇ ( φ k 2 b ′ ) ∣ φ k 2 b ′ ( 1 − b ′ ) Φ 2 b ′ ∣ ∇ Φ ∣ b ′ d x d t . \qquad+\int_0^\infty\int_{\mathbb{R}^n}\left|\nabla(\varphi_k^{2b'})\right|\varphi_k^{2b'(1-b')}\Phi^{2b'}|\nabla\Phi|^{b'}\,dxdt. + ∫ 0 ∞ ∫ R n ∇ ( φ k 2 b ′ ) φ k 2 b ′ ( 1 − b ′ ) Φ 2 b ′ ∣∇Φ ∣ b ′ d x d t . Here the constant c c c b b b κ \kappa κ k → ∞ k\to\infty k → ∞
∫ R n Φ ( ⋅ , 0 ) 2 b ′ d u 0 ≲ ∫ 0 ∞ ∫ R n ∣ ( ∂ t + Δ ) ( Φ 2 b ′ ) ∣ b ′ Φ 2 b ′ ( 1 − b ′ ) d x d t \int_{\mathbb{R}^n}\Phi(\cdot,0)^{2b'}\,du_0
\lesssim\int_0^\infty\int_{\mathbb{R}^n}\left|(\partial_t+\Delta)(\Phi^{2b'})\right|^{b'}\Phi^{2b'(1-b')}\,dxdt ∫ R n Φ ( ⋅ , 0 ) 2 b ′ d u 0 ≲ ∫ 0 ∞ ∫ R n ( ∂ t + Δ ) ( Φ 2 b ′ ) b ′ Φ 2 b ′ ( 1 − b ′ ) d x d t ≲ ∫ 0 ∞ ∫ R n ∣ ∂ t Φ + Δ Φ ∣ b ′ Φ b ′ d x d t + ∫ 0 ∞ ∫ R n ∣ ∇ Φ ∣ 2 q ′ d x d t , \lesssim\int_0^\infty\int_{\mathbb{R}^n}\left|\partial_t\Phi+\Delta\Phi\right|^{b'}\Phi^{b'}\,dxdt+\int_0^\infty\int_{\mathbb{R}^n}|\nabla\Phi|^{2q'}\,dxdt, ≲ ∫ 0 ∞ ∫ R n ∣ ∂ t Φ + ΔΦ ∣ b ′ Φ b ′ d x d t + ∫ 0 ∞ ∫ R n ∣∇Φ ∣ 2 q ′ d x d t , where we used the equality
( ∂ t + Δ ) ( Φ 2 b ′ ) = 2 b ′ Φ 2 b ′ − 1 ( ∂ t Φ + Δ Φ ) + 2 b ′ ( 2 b ′ − 1 ) Φ 2 b ′ − 2 ∣ ∇ Φ ∣ 2 . (\partial_t+\Delta)(\Phi^{2b'})=2b'\Phi^{2b'-1}(\partial_t\Phi+\Delta\Phi)+2b'(2b'-1)\Phi^{2b'-2}|\nabla\Phi|^2. ( ∂ t + Δ ) ( Φ 2 b ′ ) = 2 b ′ Φ 2 b ′ − 1 ( ∂ t Φ + ΔΦ ) + 2 b ′ ( 2 b ′ − 1 ) Φ 2 b ′ − 2 ∣∇Φ ∣ 2 . At this point, applying Gagliardo–Nirenberg interpolation inequality (see [35 , 14 ] ) in the form
∫ R n ∣ ∇ Φ ∣ 2 q ′ d x ≲ ∥ Φ ∥ L ∞ ( R n ) b ′ ∫ R n ∣ D 2 Φ ∣ b ′ d x , ∀ t ∈ R , \int_{\mathbb{R}^n}|\nabla\Phi|^{2q'}\,dx\lesssim\|\Phi\|_{L^\infty(\mathbb{R}^n)}^{b'}\int_{\mathbb{R}^n}|D^2\Phi|^{b'}\,dx,\qquad\forall t\in\mathbb{R}, ∫ R n ∣∇Φ ∣ 2 q ′ d x ≲ ∥Φ ∥ L ∞ ( R n ) b ′ ∫ R n ∣ D 2 Φ ∣ b ′ d x , ∀ t ∈ R , and the assumption 0 ≤ Φ ≤ 1 0\leq\Phi\leq1 0 ≤ Φ ≤ 1 (35) .
Inequality (35) now implies that
u 0 ( K ) ≲ cap 2 , 1 , b ′ ( K × { 0 } ) u_0(K)\lesssim\operatorname{cap}_{2,1,b'}(K\times\{0\}) u 0 ( K ) ≲ cap 2 , 1 , b ′ ( K × { 0 }) for any compact set K ⊂ R n K\subset\mathbb{R}^n K ⊂ R n Theorem 2.2 and (32) , we obtain (ii).
u 0 ∈ F ˙ q − 2 b M b 2 b , b ′ ( R n ) ⊂ B ∞ , ∞ − 2 b − 1 ( R n ) u_0 \in\dot{F}_q^{-\frac{2}{b}}\mathcal{M}_b^{\frac{2}{b},b'}(\mathbb{R}^n) \subset B_{\infty,\infty}^{-\frac{2}{b-1}}(\mathbb{R}^n) u 0 ∈ F ˙ q − b 2 M b b 2 , b ′ ( R n ) ⊂ B ∞ , ∞ − b − 1 2 ( R n ) for any 0 < q ≤ ∞ 0<q\leq\infty 0 < q ≤ ∞ (30) implies that I b 2 [ u 0 ] < + ∞ I_b^2[u_0]<+\infty I b 2 [ u 0 ] < + ∞
∫ R n 1 ∣ x 0 − y ∣ n − 2 b d u 0 ( y ) < + ∞ \int_{\mathbb{R}^n}\frac{1}{|x_0-y|^{n-\frac{2}{b}}}\,du_0(y)<+\infty ∫ R n ∣ x 0 − y ∣ n − b 2 1 d u 0 ( y ) < + ∞ for some x 0 ∈ R n x_0\in\mathbb{R}^n x 0 ∈ R n u 0 u_0 u 0 u 0 ∈ S 0 ′ ( R n ) u_0\in\mathcal{S}'_0(\mathbb{R}^n) u 0 ∈ S 0 ′ ( R n ) [4 Proposition 4.6] , u 0 u_0 u 0 (31) . □
Semilinear parabolic equations with distributional initial data
Let 1 < r < ( n + 2 ) / 2 1<r<(n+2)/2 1 < r < ( n + 2 ) /2 p > 0 p>0 p > 0 Theorem 3.3 , we define a space
Y p , r = { u ∈ L l o c p ( R + n + 1 ) : ∥ u ∥ Y p , r < + ∞ } , Y_{p,r}=\{u\in L^p_{\mathrm{loc}}(\mathbb{R}^{n+1}_+):\|u\|_{Y_{p,r}}<+\infty\}, Y p , r = { u ∈ L loc p ( R + n + 1 ) : ∥ u ∥ Y p , r < + ∞ } , where
∥ u ∥ Y p , r : = sup E ⊂ R + n + 1 ( ∬ E ∣ u ( x , t ) ∣ p d x d t cap 2 , 1 , r ( E ) ) 1 / p , \|u\|_{Y_{p,r}}:=\sup_{E\subset\mathbb{R}^{n+1}_+}\left(\frac{\iint_E|u(x,t)|^p\,dx\,dt}{\operatorname{cap}_{2,1,r}(E)}\right)^{1/p}, ∥ u ∥ Y p , r := E ⊂ R + n + 1 sup ( cap 2 , 1 , r ( E ) ∬ E ∣ u ( x , t ) ∣ p d x d t ) 1/ p , with the supremum being taken over all compact subsets E E E R + n + 1 = R n × ( 0 , ∞ ) \mathbb{R}^{n+1}_+=\mathbb{R}^n\times(0,\infty) R + n + 1 = R n × ( 0 , ∞ )
Z p , r ( R n ) : = { f ∈ S ′ ( R n ) : ∥ f ∥ Z p , r ( R n ) : = ∥ e t Δ f ∥ Y p , r < + ∞ } . Z_{p,r}(\mathbb{R}^n):=\{f\in\mathcal{S}'(\mathbb{R}^n):\|f\|_{Z_{p,r}(\mathbb{R}^n)}:=\|e^{t\Delta}f\|_{Y_{p,r}}<+\infty\}. Z p , r ( R n ) := { f ∈ S ′ ( R n ) : ∥ f ∥ Z p , r ( R n ) := ∥ e t Δ f ∥ Y p , r < + ∞ } . For b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^n) Z b , b ′ ( R n ) (24) . To show its significance, we will now demonstrate that it contains all of the spaces of initial data that were known earlier in the literature for the existence of global in time solutions. First, note that by Theorem 3.3 , for b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2
Z b , b ′ ( R n ) ∩ M + ( R n ) = F ˙ q − 2 b M b 2 b , b ′ ( R n ) ∩ M + ( R n ) Z_{b,b'}(\mathbb{R}^n)\cap\mathcal{M}^+(\mathbb{R}^n)=\dot{F}_q^{-\frac{2}{b}}\mathcal{M}_b^{\frac{2}{b},b'}(\mathbb{R}^n)\cap\mathcal{M}^+(\mathbb{R}^n) Z b , b ′ ( R n ) ∩ M + ( R n ) = F ˙ q − b 2 M b b 2 , b ′ ( R n ) ∩ M + ( R n ) for all $0<q\leq\infty.
For general signed tempered distributions, we have the following embedding results.
Theorem 4.1. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2
B ˙ b − 2 b M b 2 b , b ′ ↪ F ˙ b − 2 b M b 2 b , b ′ ( R n ) ↪ Z b , b ′ ( R n ) . (36) \dot{B}_b^{-\frac{2}{b}}\mathcal{M}_b^{\frac{2}{b},b'}\hookrightarrow\dot{F}_b^{-\frac{2}{b}}\mathcal{M}_b^{\frac{2}{b},b'}(\mathbb{R}^n)\hookrightarrow Z_{b,b'}(\mathbb{R}^n). \tag*{(36)} B ˙ b − b 2 M b b 2 , b ′ ↪ F ˙ b − b 2 M b b 2 , b ′ ( R n ) ↪ Z b , b ′ ( R n ) . (36) Similarly, for 0 < ϵ < n ( b − 1 ) / 2 − 1 0<\epsilon<n(b-1)/2-1 0 < ϵ < n ( b − 1 ) /2 − 1
B ˙ b − 2 b M b ( 1 + ϵ ) , 2 ( 1 + ϵ ) b − 1 ↪ F ˙ b − 2 b M b ( 1 + ϵ ) , 2 ( 1 + ϵ ) b − 1 ↪ Z b , b ′ ( R n ) . \dot{B}_b^{-\frac{2}{b}}\mathcal{M}^{b(1+\epsilon),\frac{2(1+\epsilon)}{b-1}}\hookrightarrow\dot{F}_b^{-\frac{2}{b}}\mathcal{M}^{b(1+\epsilon),\frac{2(1+\epsilon)}{b-1}}\hookrightarrow Z_{b,b'}(\mathbb{R}^n). B ˙ b − b 2 M b ( 1 + ϵ ) , b − 1 2 ( 1 + ϵ ) ↪ F ˙ b − b 2 M b ( 1 + ϵ ) , b − 1 2 ( 1 + ϵ ) ↪ Z b , b ′ ( R n ) . Proof. By (11) , it suffices to prove only the second embedding in (36) . Let f ∈ F ˙ b − 2 b M ˙ b 2 b , b ′ ( R n ) f \in\dot{F}_{b}^{-\frac{2}{b}}\dot{M}_{b}^{\frac{2}{b},b'}(\mathbb{R}^{n}) f ∈ F ˙ b − b 2 M ˙ b b 2 , b ′ ( R n ) E ⊂ R n × [ 0 , ∞ ) E \subset\mathbb{R}^{n} \times[0,\infty) E ⊂ R n × [ 0 , ∞ )
∬ E ∣ e t Δ f ( x ) ∣ b d x d t ≤ ∫ K ∫ 0 ∞ ∣ e t Δ f ( x ) ∣ b d t d x . \iint_{E} |e^{t\Delta}f(x)|^{b}\,dx\,dt \leq\int_{K}\int_{0}^{\infty} |e^{t\Delta}f(x)|^{b}\,dt\,dx. ∬ E ∣ e t Δ f ( x ) ∣ b d x d t ≤ ∫ K ∫ 0 ∞ ∣ e t Δ f ( x ) ∣ b d t d x . where K ⊂ R n K \subset\mathbb{R}^{n} K ⊂ R n K × { 0 } = { ( x , 0 ) : ∃ t ≥ 0 such that ( x , t ) ∈ E } K \times\{0\} = \{(x,0): \exists\,t \geq0\ \text{such that } (x,t) \in E\} K × { 0 } = {( x , 0 ) : ∃ t ≥ 0 such that ( x , t ) ∈ E } E E E { t = 0 } \{t=0\} { t = 0 } Theorem 2.16 , it follows that
∬ E ∣ e t Δ f ( x ) ∣ b d x d t ≲ ∥ f ∥ F ˙ b − 2 b M ˙ b 2 b , b ′ ( R n ) b Cap I 2 b , b ′ ( K ) \iint_{E} |e^{t\Delta}f(x)|^{b}\,dx\,dt \lesssim\|f\|_{\dot{F}_{b}^{-\frac{2}{b}}\dot{M}_{b}^{\frac{2}{b},b'}(\mathbb{R}^{n})}^{b} \operatorname{Cap}_{I_{\frac{2}{b},b'}}(K) ∬ E ∣ e t Δ f ( x ) ∣ b d x d t ≲ ∥ f ∥ F ˙ b − b 2 M ˙ b b 2 , b ′ ( R n ) b Cap I b 2 , b ′ ( K ) ≲ ∥ f ∥ F ˙ b − 2 b M ˙ b 2 b , b ′ ( R n ) b cap 2 , 1 , b ′ ( K × { 0 } ) \lesssim\|f\|_{\dot{F}_{b}^{-\frac{2}{b}}\dot{M}_{b}^{\frac{2}{b},b'}(\mathbb{R}^{n})}^{b} \operatorname{cap}_{2,1,b'}(K \times\{0\}) ≲ ∥ f ∥ F ˙ b − b 2 M ˙ b b 2 , b ′ ( R n ) b cap 2 , 1 , b ′ ( K × { 0 }) ≲ ∥ f ∥ F ˙ b − 2 b M ˙ b 2 b , b ′ ( R n ) b cap 2 , 1 , b ′ ( E ) , \lesssim\|f\|_{\dot{F}_{b}^{-\frac{2}{b}}\dot{M}_{b}^{\frac{2}{b},b'}(\mathbb{R}^{n})}^{b} \operatorname{cap}_{2,1,b'}(E), ≲ ∥ f ∥ F ˙ b − b 2 M ˙ b b 2 , b ′ ( R n ) b cap 2 , 1 , b ′ ( E ) , where we used (32) in the second inequality. In the last inequality, we used the fact that cap 2 , 1 , b ′ ( K × { 0 } ) ≲ cap 2 , 1 , b ′ ( E ) \operatorname{cap}_{2,1,b'}(K \times\{0\}) \lesssim\operatorname{cap}_{2,1,b'}(E) cap 2 , 1 , b ′ ( K × { 0 }) ≲ cap 2 , 1 , b ′ ( E )
Lemma 4.2. Let β ∈ ( 0 , n + 2 ) \beta\in(0,n+2) β ∈ ( 0 , n + 2 ) 1 < p < n + 2 β 1 < p < \frac{n+2}{\beta} 1 < p < β n + 2 E ⊂ R n + 1 E \subset\mathbb{R}^{n+1} E ⊂ R n + 1 Φ : E → R n + 1 \Phi:E \to\mathbb{R}^{n+1} Φ : E → R n + 1 L L L
∣ Φ ( z 1 ) − Φ ( z 2 ) ∣ ≤ L ∣ z 1 − z 2 ∣ |\Phi(z_{1})-\Phi(z_{2})| \leq L|z_{1}-z_{2}| ∣Φ ( z 1 ) − Φ ( z 2 ) ∣ ≤ L ∣ z 1 − z 2 ∣ for all z 1 , z 2 ∈ E z_{1},z_{2} \in E z 1 , z 2 ∈ E
Cap I β , p ( Φ ( E ) ) ≤ A L n + 2 − β p Cap I β , p ( E ) . \operatorname{Cap}_{I_{\beta,p}}(\Phi(E)) \leq AL^{n+2-\beta p}\operatorname{Cap}_{I_{\beta,p}}(E). Cap I β , p ( Φ ( E )) ≤ A L n + 2 − β p Cap I β , p ( E ) . Proof. The proof is based on the following dual definition of capacity:
Cap I β , p ( K ) 1 p = sup { μ ( K ) : μ ∈ M + ( K ) , ∥ I β [ μ ] ∥ L p ′ ( R n + 1 ) ≤ 1 } \operatorname{Cap}_{I_{\beta,p}}(K)^{\frac{1}{p}} = \sup\{\mu(K): \mu\in\mathcal{M}^{+}(K),\ \|I_{\beta}[\mu]\|_{L^{p'}(\mathbb{R}^{n+1})} \leq1\} Cap I β , p ( K ) p 1 = sup { μ ( K ) : μ ∈ M + ( K ) , ∥ I β [ μ ] ∥ L p ′ ( R n + 1 ) ≤ 1 } for any compact set K ⊂ R n + 1 K \subset\mathbb{R}^{n+1} K ⊂ R n + 1 [2 Theorem 2.5.1] . We also need the parabolic version of Wolff’s inequality:
∥ I β [ μ ] ∥ L p ′ ( R n + 1 ) p ′ ≃ ∫ R n + 1 ∫ 0 ∞ ( μ ( Q r ( x , t ) ) r n + 2 − β p ) 1 p − 1 d r r d μ ( x , t ) , \|I_{\beta}[\mu]\|_{L^{p'}(\mathbb{R}^{n+1})}^{p'} \simeq\int_{\mathbb{R}^{n+1}}\int_{0}^{\infty}\left(\frac{\mu(Q_{r}(x,t))}{r^{n+2-\beta p}}\right)^{\frac{1}{p-1}}\frac{dr}{r}\,d\mu(x,t), ∥ I β [ μ ] ∥ L p ′ ( R n + 1 ) p ′ ≃ ∫ R n + 1 ∫ 0 ∞ ( r n + 2 − β p μ ( Q r ( x , t )) ) p − 1 1 r d r d μ ( x , t ) , see [12 Theorem 2.7] . With these, the proof goes as in the proof of [2 Theorem 5.2.1] that was presented for Bessel capacities (see also [2 Corollary 5.2.3] ). □
Note that Theorem 4.1 implies that
B ˙ n b ( b − 1 ) / 2 , b − 2 / b ↪ F ˙ n b ( b − 1 ) / 2 , b − 2 / b ↪ Z b , b ′ ( R n ) , b > 1 + 2 n , \dot{B}_{nb(b-1)/2,b}^{-2/b} \hookrightarrow\dot{F}_{nb(b-1)/2,b}^{-2/b} \hookrightarrow\mathcal{Z}_{b,b'}(\mathbb{R}^{n}), \qquad b > 1+\frac{2}{n}, B ˙ nb ( b − 1 ) /2 , b − 2/ b ↪ F ˙ nb ( b − 1 ) /2 , b − 2/ b ↪ Z b , b ′ ( R n ) , b > 1 + n 2 , which can be used to answer the question of Brandolese and Cortez in [5 ] .
As another consequence of Theorem 4.1, we have the following result for locally finite signed measures, which covers the case of Morrey spaces considered in [39 ] . In fact, the case of signed measures has been considered in [34 ] .
Corollary 4.3. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 μ ∈ M ( R n ) \mu\in\mathcal{M}(\mathbb{R}^{n}) μ ∈ M ( R n )
∣ μ ∣ ( K ) ≤ A Cap I 2 b , b ′ ( K ) |\mu|(K)\le A\operatorname{Cap}_{I_{\frac{2}{b},b'}}(K) ∣ μ ∣ ( K ) ≤ A Cap I b 2 , b ′ ( K ) for all compact sets K ⊂ R n K\subset\mathbb{R}^{n} K ⊂ R n
∥ μ ∥ Z b , b ′ ( R n ) ≤ C sup K ⊂ R n { ∣ μ ∣ ( K ) Cap I 2 b , b ′ ( K ) } , \|\mu\|_{Z_{b,b'}(\mathbb{R}^{n})}\le C\sup_{K\subset\mathbb{R}^{n}}\left\{\frac{|\mu|(K)}{\operatorname{Cap}_{I_{\frac{2}{b},b'}}(K)}\right\}, ∥ μ ∥ Z b , b ′ ( R n ) ≤ C K ⊂ R n sup ⎩ ⎨ ⎧ Cap I b 2 , b ′ ( K ) ∣ μ ∣ ( K ) ⎭ ⎬ ⎫ , where the supremum is taken over all compact sets K ⊂ R n K\subset\mathbb{R}^{n} K ⊂ R n Cap I 2 b , b ′ ( K ) ≠ 0 \operatorname{Cap}_{I_{\frac{2}{b},b'}}(K)\ne0 Cap I b 2 , b ′ ( K ) = 0 1 < q ≤ n ( b − 1 ) 2 1<q\le\frac{n(b-1)}{2} 1 < q ≤ 2 n ( b − 1 )
M q , 2 q b − 1 ( R n ) ↪ Z b , b ′ ( R n ) . (37) \mathcal{M}^{q,\frac{2q}{b-1}}(\mathbb{R}^{n})\hookrightarrow Z_{b,b'}(\mathbb{R}^{n}). \tag*{(37)} M q , b − 1 2 q ( R n ) ↪ Z b , b ′ ( R n ) . (37) Proof. By Theorem 3.3 (see also (25) ), we have
∥ μ ∥ Z b , b ′ ( R n ) ≤ ∥ ∣ μ ∣ ∥ Z b , b ′ ( R n ) ≃ ∥ ∣ μ ∣ ∥ F ˙ q − 2 b M ˙ b 2 , b ′ ( R n ) ≃ sup K ⊂ R n { ∣ μ ∣ ( K ) Cap I 2 b , b ′ ( K ) } . \|\mu\|_{Z_{b,b'}(\mathbb{R}^{n})}\le\||\mu|\|_{Z_{b,b'}(\mathbb{R}^{n})}\simeq\||\mu|\|_{\dot{F}_{q}^{-\frac{2}{b}}\dot{M}_{b}^{2,b'}(\mathbb{R}^{n})}\simeq\sup_{K\subset\mathbb{R}^{n}}\left\{\frac{|\mu|(K)}{\operatorname{Cap}_{I_{\frac{2}{b},b'}}(K)}\right\}. ∥ μ ∥ Z b , b ′ ( R n ) ≤ ∥∣ μ ∣ ∥ Z b , b ′ ( R n ) ≃ ∥∣ μ ∣ ∥ F ˙ q − b 2 M ˙ b 2 , b ′ ( R n ) ≃ K ⊂ R n sup ⎩ ⎨ ⎧ Cap I b 2 , b ′ ( K ) ∣ μ ∣ ( K ) ⎭ ⎬ ⎫ . The embedding (37) follows from the above inequality and Fefferman-Phong inequality.
We next show the relation of Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^{n}) Z b , b ′ ( R n )
Theorem 4.4. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 ϵ ≥ 0 \epsilon\ge0 ϵ ≥ 0
T b , ϵ ( R n ) : = { f ∈ S ′ ( R n ) : ∥ f ∥ T b , ϵ ( R n ) < + ∞ } , T_{b,\epsilon}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}'(\mathbb{R}^{n}):\|f\|_{T_{b,\epsilon}(\mathbb{R}^{n})}<+\infty\right\}, T b , ϵ ( R n ) := { f ∈ S ′ ( R n ) : ∥ f ∥ T b , ϵ ( R n ) < + ∞ } , where
∥ f ∥ T b , ϵ ( R n ) : = ( sup x ∈ R n , r > 0 r 2 b ( 1 + ϵ ) b − 1 − n − 2 ∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ) 1 b ( 1 + ϵ ) . \|f\|_{T_{b,\epsilon}(\mathbb{R}^{n})}:=\left(\sup_{x\in\mathbb{R}^{n},\,r>0}r^{\frac{2b(1+\epsilon)}{b-1}-n-2}\int_{0}^{r^{2}}\int_{B_{r}(x)}|e^{t\Delta}f|^{b(1+\epsilon)}\,dy\,dt\right)^{\frac{1}{b(1+\epsilon)}}. ∥ f ∥ T b , ϵ ( R n ) := ( x ∈ R n , r > 0 sup r b − 1 2 b ( 1 + ϵ ) − n − 2 ∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ) b ( 1 + ϵ ) 1 . Then we have
T b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) ↪ T b , 0 ( R n ) , ϵ > 0. T_{b,\epsilon}(\mathbb{R}^{n})\hookrightarrow Z_{b,b'}(\mathbb{R}^{n})\hookrightarrow T_{b,0}(\mathbb{R}^{n}),\qquad\epsilon>0. T b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) ↪ T b , 0 ( R n ) , ϵ > 0. Proof. The embedding Z b , b ′ ( R n ) ↪ T b , 0 ( R n ) Z_{b,b'}(\mathbb{R}^{n})\hookrightarrow T_{b,0}(\mathbb{R}^{n}) Z b , b ′ ( R n ) ↪ T b , 0 ( R n ) cap 1 , 2 , b ′ ( Q r ) = c r n + 2 − 2 b ′ \operatorname{cap}_{1,2,b'}(Q_{r})=cr^{n+2-2b'} cap 1 , 2 , b ′ ( Q r ) = c r n + 2 − 2 b ′ r > 0 r>0 r > 0 ϵ ≥ 0 \epsilon\ge0 ϵ ≥ 0
∥ e t Δ f ∥ L ∞ ( R n ) ≲ ∥ f ∥ T b , ϵ ( R n ) t − 1 b − 1 , t > 0. (38) \|e^{t\Delta}f\|_{L^{\infty}(\mathbb{R}^{n})}\lesssim\|f\|_{T_{b,\epsilon}(\mathbb{R}^{n})}t^{-\frac{1}{b-1}},\qquad t>0. \tag*{(38)} ∥ e t Δ f ∥ L ∞ ( R n ) ≲ ∥ f ∥ T b , ϵ ( R n ) t − b − 1 1 , t > 0. (38) For this, see Lemma 4.6 below. To prove the embedding T b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) T_{b,\epsilon}(\mathbb{R}^{n})\hookrightarrow Z_{b,b'}(\mathbb{R}^{n}) T b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) 2 b ( 1 + ϵ ) b − 1 ≤ n + 2 \frac{2b(1+\epsilon)}{b-1}\le n+2 b − 1 2 b ( 1 + ϵ ) ≤ n + 2
∫ t 0 t 0 + r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ ∥ f ∥ T b , ϵ ( R n ) b ( 1 + ϵ ) r n + 2 − 2 b ( 1 + ϵ ) b − 1 . (39) \int_{t_{0}}^{t_{0}+r^{2}}\int_{B_{r}(x)}|e^{t\Delta}f|^{b(1+\epsilon)}\,dy\,dt\lesssim\|f\|_{T_{b,\epsilon}(\mathbb{R}^{n})}^{b(1+\epsilon)}r^{n+2-\frac{2b(1+\epsilon)}{b-1}}. \tag*{(39)} ∫ t 0 t 0 + r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ ∥ f ∥ T b , ϵ ( R n ) b ( 1 + ϵ ) r n + 2 − b − 1 2 b ( 1 + ϵ ) . (39) for all t 0 ≥ 0 t_0 \ge0 t 0 ≥ 0 r > 0 r > 0 r > 0 x ∈ R n x \in\mathbb{R}^n x ∈ R n ϵ ≥ 0 \epsilon\ge0 ϵ ≥ 0 t 0 ≤ r 2 t_0 \le r^2 t 0 ≤ r 2
∫ t 0 t 0 + r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ ∫ 0 t 0 + r 2 ∫ B t 0 + r 2 ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t \int_{t_0}^{t_0+r^2}\int_{B_r(x)} \lvert e^{t\Delta}f\rvert^{b(1+\epsilon)}\,dy\,dt \lesssim\int_0^{t_0+r^2}\int_{B_{\sqrt{t_0+r^2}}(x)} \lvert e^{t\Delta}f\rvert^{b(1+\epsilon)}\,dy\,dt ∫ t 0 t 0 + r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ ∫ 0 t 0 + r 2 ∫ B t 0 + r 2 ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) ( t 0 + r 2 ) n + 2 − 2 b ( 1 + ϵ ) b − 1 \lesssim\lVert f\rVert_{\dot{T}_{b,\epsilon}(\mathbb{R}^n)}^{b(1+\epsilon)}\left(\sqrt{t_0+r^2}\right)^{n+2-\frac{2b(1+\epsilon)}{b-1}} ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) ( t 0 + r 2 ) n + 2 − b − 1 2 b ( 1 + ϵ ) ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) r n + 2 − 2 b ( 1 + ϵ ) b − 1 . \lesssim\lVert f\rVert_{\dot{T}_{b,\epsilon}(\mathbb{R}^n)}^{b(1+\epsilon)}r^{n+2-\frac{2b(1+\epsilon)}{b-1}}. ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) r n + 2 − b − 1 2 b ( 1 + ϵ ) . On the other hand, if t 0 > r 2 t_0 > r^2 t 0 > r 2 (38) ,
∫ t 0 t 0 + r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ r n ∫ t 0 t 0 + r 2 ∥ e t Δ f ∥ L ∞ ( R n ) b ( 1 + ϵ ) d t \int_{t_0}^{t_0+r^2}\int_{B_r(x)} \lvert e^{t\Delta}f\rvert^{b(1+\epsilon)}\,dy\,dt \lesssim r^n\int_{t_0}^{t_0+r^2}\lVert e^{t\Delta}f\rVert_{L^\infty(\mathbb{R}^n)}^{b(1+\epsilon)}\,dt ∫ t 0 t 0 + r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ r n ∫ t 0 t 0 + r 2 ∥ e t Δ f ∥ L ∞ ( R n ) b ( 1 + ϵ ) d t ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) r n ∫ t 0 t 0 + r 2 t − b ( 1 + ϵ ) b − 1 d t \lesssim\lVert f\rVert_{\dot{T}_{b,\epsilon}(\mathbb{R}^n)}^{b(1+\epsilon)}r^n\int_{t_0}^{t_0+r^2}t^{-\frac{b(1+\epsilon)}{b-1}}\,dt ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) r n ∫ t 0 t 0 + r 2 t − b − 1 b ( 1 + ϵ ) d t ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) r n + 2 − 2 b ( 1 + ϵ ) b − 1 . \lesssim\lVert f\rVert_{\dot{T}_{b,\epsilon}(\mathbb{R}^n)}^{b(1+\epsilon)}r^{n+2-\frac{2b(1+\epsilon)}{b-1}}. ≲ ∥ f ∥ T ˙ b , ϵ ( R n ) b ( 1 + ϵ ) r n + 2 − b − 1 2 b ( 1 + ϵ ) . Thus we obtain the claim (39) as desired. Note that (39) , with ϵ > 0 \epsilon> 0 ϵ > 0 [38 Theorem 4] we arrive at the embedding T ˙ b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) \dot{T}_{b,\epsilon}(\mathbb{R}^n) \hookrightarrow Z_{b,b'}(\mathbb{R}^n) T ˙ b , ϵ ( R n ) ↪ Z b , b ′ ( R n ) ϵ > 0 \epsilon> 0 ϵ > 0
In [22 ] , Kozono and Yamayaki used the Besov-Morrey space B ˙ ∞ n p − 2 b − 1 M q , n q p \dot{B}^{\frac{n}{p}-\frac{2}{b-1}}_{\infty}\mathcal{M}^{q,\frac{nq}{p}} B ˙ ∞ p n − b − 1 2 M q , p n q p p p q q q (24) . We now show that this space is also contained in Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^n) Z b , b ′ ( R n )
Corollary 4.5. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 p , q p,q p , q b ≤ q ≤ p b\le q\le p b ≤ q ≤ p n ( b − 1 ) / 2 < p < n b ( b − 1 ) / 2 n(b-1)/2<p<nb(b-1)/2 n ( b − 1 ) /2 < p < nb ( b − 1 ) /2
Then we have
F ˙ ∞ n p − 2 b − 1 M q , n q p ↪ B ˙ ∞ n p − 2 b − 1 M q , n q p ↪ Z b , b ′ ( R n ) . \dot{F}^{\frac{n}{p}-\frac{2}{b-1}}_{\infty}\mathcal{M}^{q,\frac{nq}{p}} \hookrightarrow\dot{B}^{\frac{n}{p}-\frac{2}{b-1}}_{\infty}\mathcal{M}^{q,\frac{nq}{p}} \hookrightarrow Z_{b,b'}(\mathbb{R}^n). F ˙ ∞ p n − b − 1 2 M q , p n q ↪ B ˙ ∞ p n − b − 1 2 M q , p n q ↪ Z b , b ′ ( R n ) . Proof. The first embedding is obvious. For the second embedding, note that for any b < q ≤ p b<q\le p b < q ≤ p
M q , n q p ↪ M b , n b p \mathcal{M}^{q,\frac{nq}{p}} \hookrightarrow\mathcal{M}^{b,\frac{nb}{p}} M q , p n q ↪ M b , p nb by Hölder’s inequality. Thus, it suffices to consider only the endpoint case q = b q=b q = b [39 Proposition 4.1] : For any 1 ≤ p 1 ≤ p 2 < ∞ 1\le p_1\le p_2<\infty 1 ≤ p 1 ≤ p 2 < ∞ λ ∈ ( 0 , n ] \lambda\in(0,n] λ ∈ ( 0 , n ]
∥ e t Δ f ∥ M p 2 , λ ≲ t − λ 2 ( 1 p 1 − 1 p 2 ) ∥ e t Δ f ∥ M p 1 , λ , t > 0. \lVert e^{t\Delta}f\rVert_{\mathcal{M}^{p_2,\lambda}}\lesssim t^{-\frac{\lambda}{2}\left(\frac{1}{p_1}-\frac{1}{p_2}\right)}\lVert e^{t\Delta}f\rVert_{\mathcal{M}^{p_1,\lambda}},\qquad t>0. ∥ e t Δ f ∥ M p 2 , λ ≲ t − 2 λ ( p 1 1 − p 2 1 ) ∥ e t Δ f ∥ M p 1 , λ , t > 0. Thus, for b ≤ p < n b ( b − 1 ) / 2 b\le p<nb(b-1)/2 b ≤ p < nb ( b − 1 ) /2 f ∈ B ˙ ∞ n p − 2 b − 1 M b , n b p f\in\dot{B}^{\frac{n}{p}-\frac{2}{b-1}}_{\infty}\mathcal{M}^{b,\frac{nb}{p}} f ∈ B ˙ ∞ p n − b − 1 2 M b , p nb
N f = ∥ f ∥ B ˙ ∞ n p − 2 b − 1 M b , n b p , N_f=\lVert f\rVert_{\dot{B}^{\frac{n}{p}-\frac{2}{b-1}}_{\infty}\mathcal{M}^{b,\frac{nb}{p}}}, N f = ∥ f ∥ B ˙ ∞ p n − b − 1 2 M b , p nb , then
∥ e t Δ f ∥ M b ( 1 + ϵ ) , n b p ≲ t − n b 2 p ( 1 b − 1 b ( 1 + ϵ ) ) ∥ e t Δ f ∥ M b , n b p \|e^{t\Delta}f\|_{\mathcal{M}^{b(1+\epsilon),\frac{nb}{p}}}
\lesssim t^{-\frac{nb}{2p}\left(\frac{1}{b}-\frac{1}{b(1+\epsilon)}\right)}
\|e^{t\Delta}f\|_{\mathcal{M}^{b,\frac{nb}{p}}} ∥ e t Δ f ∥ M b ( 1 + ϵ ) , p nb ≲ t − 2 p nb ( b 1 − b ( 1 + ϵ ) 1 ) ∥ e t Δ f ∥ M b , p nb ≲ N f t − n b 2 p ( 1 b − 1 b ( 1 + ϵ ) ) − 1 b − 1 + n 2 p \lesssim N_f t^{-\frac{nb}{2p}\left(\frac{1}{b}-\frac{1}{b(1+\epsilon)}\right)-\frac{1}{b-1}+\frac{n}{2p}} ≲ N f t − 2 p nb ( b 1 − b ( 1 + ϵ ) 1 ) − b − 1 1 + 2 p n ≲ N f t − 1 b − 1 + n 2 p ( 1 + ϵ ) , \lesssim N_f t^{-\frac{1}{b-1}+\frac{n}{2p(1+\epsilon)}}, ≲ N f t − b − 1 1 + 2 p ( 1 + ϵ ) n , where in the second inequality we used a characterization of Besov-Morrey space of order s = n / p − 2 / ( b − 1 ) < 0 s=n/p-2/(b-1)<0 s = n / p − 2/ ( b − 1 ) < 0 [28 Proposition 2.22 and Equ. (2.57)] . It follows that for any x ∈ R n x\in\mathbb{R}^{n} x ∈ R n r > 0 r>0 r > 0
∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ ( N f ) b ( 1 + ϵ ) r n − n b p ∫ 0 r 2 t − b ( 1 + ϵ ) b − 1 + n b 2 p d t . \int_{0}^{r^{2}}\int_{B_{r}(x)}|e^{t\Delta}f|^{b(1+\epsilon)}\,dy\,dt
\lesssim(N_f)^{b(1+\epsilon)}r^{n-\frac{nb}{p}}\int_{0}^{r^{2}}t^{-\frac{b(1+\epsilon)}{b-1}+\frac{nb}{2p}}\,dt. ∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d t ≲ ( N f ) b ( 1 + ϵ ) r n − p nb ∫ 0 r 2 t − b − 1 b ( 1 + ϵ ) + 2 p nb d t . Note that, since p < n b ( b − 1 ) / 2 p<nb(b-1)/2 p < nb ( b − 1 ) /2
− b ( 1 + ϵ ) b − 1 + n b 2 p > − 1 , -\frac{b(1+\epsilon)}{b-1}+\frac{nb}{2p}>-1, − b − 1 b ( 1 + ϵ ) + 2 p nb > − 1 , provided we choose
0 < ϵ < ( 1 + n b 2 p ) b − 1 b − 1. 0<\epsilon<\left(1+\frac{nb}{2p}\right)\frac{b-1}{b}-1. 0 < ϵ < ( 1 + 2 p nb ) b b − 1 − 1. Thus, for this choice of ϵ \epsilon ϵ
∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d s ≲ ( N f ) b ( 1 + ϵ ) r n − n b p r − 2 b ( 1 + ϵ ) b − 1 + n b p + 2 \int_{0}^{r^{2}}\int_{B_{r}(x)}|e^{t\Delta}f|^{b(1+\epsilon)}\,dy\,ds
\lesssim(N_f)^{b(1+\epsilon)}r^{n-\frac{nb}{p}}r^{-\frac{2b(1+\epsilon)}{b-1}+\frac{nb}{p}+2} ∫ 0 r 2 ∫ B r ( x ) ∣ e t Δ f ∣ b ( 1 + ϵ ) d y d s ≲ ( N f ) b ( 1 + ϵ ) r n − p nb r − b − 1 2 b ( 1 + ϵ ) + p nb + 2 ≲ ( N f ) b ( 1 + ϵ ) r n + 2 − 2 b ( 1 + ϵ ) b − 1 , \lesssim(N_f)^{b(1+\epsilon)}r^{n+2-\frac{2b(1+\epsilon)}{b-1}}, ≲ ( N f ) b ( 1 + ϵ ) r n + 2 − b − 1 2 b ( 1 + ϵ ) , which by Theorem 4.4 yields the desired embedding. □
Lemma 4.6. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 Z b , b ′ ( R n ) ↪ B ˙ ∞ , ∞ − 2 b − 1 ( R n ) ∩ S ′ ( R n ) Z_{b,b'}(\mathbb{R}^{n})\hookrightarrow\dot{B}_{\infty,\infty}^{-\frac{2}{b-1}}(\mathbb{R}^{n})\cap\mathcal{S}'(\mathbb{R}^{n}) Z b , b ′ ( R n ) ↪ B ˙ ∞ , ∞ − b − 1 2 ( R n ) ∩ S ′ ( R n )
∥ e t Δ f ∥ L ∞ ≲ t − 1 b − 1 ∥ f ∥ T b , 0 ( R n ) ≲ t − 1 b − 1 ∥ f ∥ Z b , b ′ ( R n ) , ∀ t > 0. (40) \|e^{t\Delta}f\|_{L^{\infty}}\lesssim t^{-\frac{1}{b-1}}\|f\|_{T_{b,0}(\mathbb{R}^{n})}\lesssim t^{-\frac{1}{b-1}}\|f\|_{Z_{b,b'}(\mathbb{R}^{n})},\qquad\forall\,t>0. \tag*{(40)} ∥ e t Δ f ∥ L ∞ ≲ t − b − 1 1 ∥ f ∥ T b , 0 ( R n ) ≲ t − b − 1 1 ∥ f ∥ Z b , b ′ ( R n ) , ∀ t > 0. (40) In fact, for any R > 0 R>0 R > 0
sup 0 < t < R t 1 b − 1 ∥ e t Δ f ∥ L ∞ ≲ ∥ f ∥ T b , 0 n R ( R n ) , \sup_{0<t<R}t^{\frac{1}{b-1}}\|e^{t\Delta}f\|_{L^{\infty}}
\lesssim\|f\|_{T_{b,0}^{\sqrt{n}R}(\mathbb{R}^{n})}, 0 < t < R sup t b − 1 1 ∥ e t Δ f ∥ L ∞ ≲ ∥ f ∥ T b , 0 n R ( R n ) , where T b , 0 M ( R n ) T_{b,0}^{M}(\mathbb{R}^{n}) T b , 0 M ( R n ) M > 0 M>0 M > 0 T b , 0 ( R n ) T_{b,0}(\mathbb{R}^{n}) T b , 0 ( R n )
∥ f ∥ T b , 0 M ( R n ) : = ( sup 0 < r < M , x ∈ R n r n + 2 − 2 b ′ ∫ 0 r 2 ∫ B r ( x ) ∣ e s Δ f ∣ b d y d s ) 1 b . \|f\|_{T_{b,0}^{M}(\mathbb{R}^{n})}:=
\left(\sup_{\substack{0<r<M,\,x\in\mathbb{R}^{n}}}
r^{n+2-2b'}\int_{0}^{r^{2}}\int_{B_{r}(x)}|e^{s\Delta}f|^{b}\,dy\,ds\right)^{\frac{1}{b}}. ∥ f ∥ T b , 0 M ( R n ) := ( 0 < r < M , x ∈ R n sup r n + 2 − 2 b ′ ∫ 0 r 2 ∫ B r ( x ) ∣ e s Δ f ∣ b d y d s ) b 1 . Proof. Let ( x , t ) ∈ R + n + 1 (x,t)\in\mathbb{R}^{n+1}_+ ( x , t ) ∈ R + n + 1 τ ∈ ( 0 , t / 2 ) \tau\in(0,t/2) τ ∈ ( 0 , t /2 ) e t Δ f ( x ) = e ( t − τ ) Δ [ e τ Δ f ] ( x ) e^{t\Delta}f(x)=e^{(t-\tau)\Delta}[e^{\tau\Delta}f](x) e t Δ f ( x ) = e ( t − τ ) Δ [ e τ Δ f ] ( x )
∣ e t Δ f ( x ) ∣ = ∣ 2 t ∫ 0 t / 2 ∫ R n e − ∣ x − y ∣ 2 4 ( t − τ ) ( 4 π ( t − τ ) ) n / 2 e τ Δ f ( y ) d y d τ ∣ ≤ { 2 t ∫ 0 t / 2 ∫ R n e − ∣ x − y ∣ 2 4 ( t − τ ) ( 4 π ( t − τ ) ) n / 2 ∣ e τ Δ f ( y ) ∣ b d y d τ } 1 b . \begin{aligned}
\left|e^{t\Delta}f(x)\right|=\left|\frac{2}{t}\int_0^{t/2}\int_{\mathbb{R}^n}\frac{e^{-\frac{|x-y|^2}{4(t-\tau)}}}{(4\pi(t-\tau))^{n/2}}e^{\tau\Delta}f(y)\,dy\,d\tau\right| \\
&\leq\left\{\frac{2}{t}\int_0^{t/2}\int_{\mathbb{R}^n}\frac{e^{-\frac{|x-y|^2}{4(t-\tau)}}}{(4\pi(t-\tau))^{n/2}}\left|e^{\tau\Delta}f(y)\right|^b\,dy\,d\tau\right\}^{\frac{1}{b}}.
\end{aligned} e t Δ f ( x ) = t 2 ∫ 0 t /2 ∫ R n ( 4 π ( t − τ ) ) n /2 e − 4 ( t − τ ) ∣ x − y ∣ 2 e τ Δ f ( y ) d y d τ ≤ ⎩ ⎨ ⎧ t 2 ∫ 0 t /2 ∫ R n ( 4 π ( t − τ ) ) n /2 e − 4 ( t − τ ) ∣ x − y ∣ 2 e τ Δ f ( y ) b d y d τ ⎭ ⎬ ⎫ b 1 . Observe that if τ ∈ ( 0 , t / 2 ) \tau\in(0,t/2) τ ∈ ( 0 , t /2 ) x − y t ∈ k + [ 0 , 1 ] n \frac{x-y}{\sqrt{t}}\in k+[0,1]^n t x − y ∈ k + [ 0 , 1 ] n k ∈ Z n k\in\mathbb{Z}^n k ∈ Z n
e − ∣ x − y ∣ 2 4 ( t − τ ) ( 4 π ( t − τ ) ) n / 2 ≤ e − ∣ x − y ∣ 2 4 t ( 4 π ( t − τ ) ) n / 2 ≤ 1 t n / 2 C ( n , N ) ( 1 + ∣ k ∣ ) N , ∀ N > n . (41) \frac{e^{-\frac{|x-y|^2}{4(t-\tau)}}}{(4\pi(t-\tau))^{n/2}}\leq\frac{e^{-\frac{|x-y|^2}{4t}}}{(4\pi(t-\tau))^{n/2}}\leq\frac{1}{t^{n/2}}\frac{C(n,N)}{(1+|k|)^N},\qquad\forall N>n. \tag*{(41)} ( 4 π ( t − τ ) ) n /2 e − 4 ( t − τ ) ∣ x − y ∣ 2 ≤ ( 4 π ( t − τ ) ) n /2 e − 4 t ∣ x − y ∣ 2 ≤ t n /2 1 ( 1 + ∣ k ∣ ) N C ( n , N ) , ∀ N > n . (41) Thus, letting N = n + 1 N=n+1 N = n + 1
∥ e t Δ f ∥ L ∞ ≲ { ∑ k ∈ Z n t − n + 2 2 ( 1 + ∣ k ∣ ) n + 1 ∫ 0 t / 2 ∫ x − y t ∈ k + [ 0 , 1 ] n ∣ e τ Δ f ( y ) ∣ b d y d τ } 1 b \left\|e^{t\Delta}f\right\|_{L^\infty}\lesssim\left\{\sum_{k\in\mathbb{Z}^n}\frac{t^{-\frac{n+2}{2}}}{(1+|k|)^{n+1}}\int_0^{t/2}\int_{\frac{x-y}{\sqrt{t}}\in k+[0,1]^n}\left|e^{\tau\Delta}f(y)\right|^b\,dy\,d\tau\right\}^{\frac{1}{b}} e t Δ f L ∞ ≲ { k ∈ Z n ∑ ( 1 + ∣ k ∣ ) n + 1 t − 2 n + 2 ∫ 0 t /2 ∫ t x − y ∈ k + [ 0 , 1 ] n e τ Δ f ( y ) b d y d τ } b 1 ≲ { ∑ k ∈ Z n t − n + 2 2 ( 1 + ∣ k ∣ ) n + 1 t ( n + 2 − 2 b ′ ) / 2 } 1 b ∥ f ∥ T b , 0 n R ( R n ) \lesssim\left\{\sum_{k\in\mathbb{Z}^n}\frac{t^{-\frac{n+2}{2}}}{(1+|k|)^{n+1}}t^{(n+2-2b')/2}\right\}^{\frac{1}{b}}\|f\|_{T^{\sqrt{n}R}_{b,0}(\mathbb{R}^n)} ≲ { k ∈ Z n ∑ ( 1 + ∣ k ∣ ) n + 1 t − 2 n + 2 t ( n + 2 − 2 b ′ ) /2 } b 1 ∥ f ∥ T b , 0 n R ( R n ) ≲ t − 1 b − 1 ∥ f ∥ T b , 0 ( R n ) \lesssim t^{-\frac{1}{b-1}}\|f\|_{T_{b,0}(\mathbb{R}^n)} ≲ t − b − 1 1 ∥ f ∥ T b , 0 ( R n ) ≲ t − 1 b − 1 ∥ f ∥ Z b , b ′ ( R n ) . \lesssim t^{-\frac{1}{b-1}}\|f\|_{Z_{b,b'}(\mathbb{R}^n)}. ≲ t − b − 1 1 ∥ f ∥ Z b , b ′ ( R n ) . In order to prove that f ∈ S 0 ′ ( R n ) f\in S'_0(\mathbb{R}^n) f ∈ S 0 ′ ( R n ) ϕ ∈ C c ∞ ( R n ) \phi\in C^\infty_c(\mathbb{R}^n) ϕ ∈ C c ∞ ( R n ) t > 0 t>0 t > 0
F − 1 ( ϕ ( t ⋅ ) F f ) = F − 1 ( ϕ ( t ⋅ ) e t 2 ∣ ⋅ ∣ 2 e − t 2 ∣ ⋅ ∣ 2 F f ) = F − 1 ( ϕ ( t ⋅ ) e t 2 ∣ ⋅ ∣ 2 ) ∗ F − 1 ( e − t 2 ∣ ⋅ ∣ 2 F f ) = t − n F − 1 ( ϕ ( ⋅ ) e ∣ ⋅ ∣ 2 ) ( ⋅ / t ) ∗ F − 1 ( e − t 2 ∣ ⋅ ∣ 2 F f ) . \begin{aligned}
\mathcal{F}^{-1}(\phi(t\cdot)\mathcal{F}f)=\mathcal{F}^{-1}\left(\phi(t\cdot)e^{t^2|\cdot|^2}e^{-t^2|\cdot|^2}\mathcal{F}f\right) \\
&=\mathcal{F}^{-1}\left(\phi(t\cdot)e^{t^2|\cdot|^2}\right)*\mathcal{F}^{-1}\left(e^{-t^2|\cdot|^2}\mathcal{F}f\right) \\
&=t^{-n}\mathcal{F}^{-1}\left(\phi(\cdot)e^{|\cdot|^2}\right)(\cdot/t)*\mathcal{F}^{-1}\left(e^{-t^2|\cdot|^2}\mathcal{F}f\right).
\end{aligned} F − 1 ( ϕ ( t ⋅ ) F f ) = F − 1 ( ϕ ( t ⋅ ) e t 2 ∣ ⋅ ∣ 2 e − t 2 ∣ ⋅ ∣ 2 F f ) = F − 1 ( ϕ ( t ⋅ ) e t 2 ∣ ⋅ ∣ 2 ) ∗ F − 1 ( e − t 2 ∣ ⋅ ∣ 2 F f ) = t − n F − 1 ( ϕ ( ⋅ ) e ∣ ⋅ ∣ 2 ) ( ⋅ / t ) ∗ F − 1 ( e − t 2 ∣ ⋅ ∣ 2 F f ) . Thus, it follows from (40) that
∥ F − 1 ( ϕ ( t ⋅ ) F f ) ∥ L ∞ = ∥ F − 1 ( ϕ ( ⋅ ) e ∣ ⋅ ∣ 2 ) ∥ L 1 ∥ F − 1 ( e − t 2 ∣ ⋅ ∣ 2 F f ) ∥ L ∞ = C ( ϕ ) ∥ e t 2 Δ f ∥ L ∞ ≤ C ( ϕ ) C t − 2 b − 1 ∥ f ∥ Z b , b ′ ( R n ) \begin{aligned}
\left\|\mathcal{F}^{-1}(\phi(t\cdot)\mathcal{F}f)\right\|_{L^\infty}=\left\|\mathcal{F}^{-1}(\phi(\cdot)e^{|\cdot|^2})\right\|_{L^1}\left\|\mathcal{F}^{-1}(e^{-t^2|\cdot|^2}\mathcal{F}f)\right\|_{L^\infty} \\
&=C(\phi)\left\|e^{t^2\Delta}f\right\|_{L^\infty} \\
&\leq C(\phi)Ct^{-\frac{2}{b-1}}\|f\|_{Z_{b,b'}(\mathbb{R}^n)}
\end{aligned} F − 1 ( ϕ ( t ⋅ ) F f ) L ∞ = F − 1 ( ϕ ( ⋅ ) e ∣ ⋅ ∣ 2 ) L 1 F − 1 ( e − t 2 ∣ ⋅ ∣ 2 F f ) L ∞ = C ( ϕ ) e t 2 Δ f L ∞ ≤ C ( ϕ ) C t − b − 1 2 ∥ f ∥ Z b , b ′ ( R n ) as t → ∞ t\to\infty t → ∞ f = ∑ j ∈ Z Δ ˙ j f f=\sum_{j\in\mathbb{Z}}\dot{\Delta}_j f f = ∑ j ∈ Z Δ ˙ j f S ′ ( R n ) S'(\mathbb{R}^n) S ′ ( R n ) S 0 ′ ( R n ) S'_0(\mathbb{R}^n) S 0 ′ ( R n ) Proposition 2.12 . ∎
We now define a space
X = { u ∈ L l o c b ( R + n + 1 ) : ∥ u ∥ X < + ∞ } . X=\left\{u\in L_{\mathrm{loc}}^{b}(\mathbb{R}^{n+1}_{+}):\|u\|_{X}<+\infty\right\}. X = { u ∈ L loc b ( R + n + 1 ) : ∥ u ∥ X < + ∞ } . where
∥ u ∥ X : = sup t > 0 t 1 b − 1 ∥ u ( ⋅ , t ) ∥ L ∞ + ∥ u ∥ Y b , b ′ . \|u\|_{X}:=\sup_{t>0}t^{\frac{1}{b-1}}\|u(\cdot,t)\|_{L^{\infty}}+\|u\|_{Y_{b,b'}}. ∥ u ∥ X := t > 0 sup t b − 1 1 ∥ u ( ⋅ , t ) ∥ L ∞ + ∥ u ∥ Y b , b ′ . The space X X X (24) with initial data u 0 ∈ Z b , b ′ ( R n ) u_{0}\in Z_{b,b'}(\mathbb{R}^{n}) u 0 ∈ Z b , b ′ ( R n )
Lemma 4.7. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 u ∈ X u\in X u ∈ X ( x , t ) ∈ R + n + 1 (x,t)\in\mathbb{R}^{n+1}_{+} ( x , t ) ∈ R + n + 1
T ( u ) ( x , t ) : = H ∗ ( ∣ u ∣ b − 1 u χ ( 0 , ∞ ) ( ⋅ ) ) ( x , t ) T(u)(x,t):=\mathsf{H}*\left(|u|^{b-1}u\chi_{(0,\infty)}(\cdot)\right)(x,t) T ( u ) ( x , t ) := H ∗ ( ∣ u ∣ b − 1 u χ ( 0 , ∞ ) ( ⋅ ) ) ( x , t ) = ∫ 0 t e ( t − τ ) Δ [ ∣ u ( τ ) ∣ b − 1 u ( τ ) ] d τ . =\int_{0}^{t}e^{(t-\tau)\Delta}\left[|u(\tau)|^{b-1}u(\tau)\right]\,d\tau. = ∫ 0 t e ( t − τ ) Δ [ ∣ u ( τ ) ∣ b − 1 u ( τ ) ] d τ . Then we have
∥ T ( u ) ∥ X ≲ ∥ u ∥ X b , ∀ u ∈ X . \|T(u)\|_{X}\lesssim\|u\|_{X}^{b},\qquad\forall u\in X. ∥ T ( u ) ∥ X ≲ ∥ u ∥ X b , ∀ u ∈ X . This result also holds if ∣ u ∣ b − 1 u |u|^{b-1}u ∣ u ∣ b − 1 u ∣ u ∣ b |u|^{b} ∣ u ∣ b T T T R > 0 R>0 R > 0
sup 0 < t < R t 1 b − 1 ∥ T ( u ) ( ⋅ , t ) ∥ L ∞ ( R n ) ≲ ( sup 0 < s < R s 1 b − 1 ∥ u ( ⋅ , s ) ∥ L ∞ ( R n ) + ∥ u ∥ Y b b ~ n R ) b , \sup_{0<t<R}t^{\frac{1}{b-1}}\|T(u)(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{n})}\lesssim\left(\sup_{0<s<R}s^{\frac{1}{b-1}}\|u(\cdot,s)\|_{L^{\infty}(\mathbb{R}^{n})}+\|u\|_{Y_{b}^{\widetilde{b}\sqrt{nR}}}\right)^{b}, 0 < t < R sup t b − 1 1 ∥ T ( u ) ( ⋅ , t ) ∥ L ∞ ( R n ) ≲ ( 0 < s < R sup s b − 1 1 ∥ u ( ⋅ , s ) ∥ L ∞ ( R n ) + ∥ u ∥ Y b b n R ) b , where we define
∥ u ∥ Y b M : = sup 0 < r < M , x ∈ R n ( r n + 2 − 2 b ′ ∫ 0 r 2 ∫ B r ( x ) ∣ u ( y , s ) ∣ b d y d s ) 1 b , M > 0. \|u\|_{Y_{b}^{M}}:=\sup_{0<r<M,\,x\in\mathbb{R}^{n}}\left(r^{n+2-2b'}\int_{0}^{r^{2}}\int_{B_{r}(x)}|u(y,s)|^{b}\,dy\,ds\right)^{\frac{1}{b}},\qquad M>0. ∥ u ∥ Y b M := 0 < r < M , x ∈ R n sup ( r n + 2 − 2 b ′ ∫ 0 r 2 ∫ B r ( x ) ∣ u ( y , s ) ∣ b d y d s ) b 1 , M > 0. Proof. We write, for ( x , t ) ∈ R + n + 1 (x,t)\in\mathbb{R}^{n+1}_{+} ( x , t ) ∈ R + n + 1
T ( u ) ( x , t ) = ∫ 0 t 2 ∫ R n e − ∣ x − y ∣ 2 4 ( t − τ ) ( 4 π ( t − τ ) ) n 2 ∣ u ( y , τ ) ∣ b − 1 u ( y , τ ) d y d τ + ∫ t 2 t ∫ R n e − ∣ x − y ∣ 2 4 ( t − τ ) ( 4 π ( t − τ ) ) n 2 ∣ u ( y , τ ) ∣ b − 1 u ( y , τ ) d y d τ = : T 1 ( u ) ( x , t ) + T 2 ( u ) ( x , t ) . \begin{aligned}
T(u)(x,t)
&=\int_{0}^{\frac{t}{2}}\int_{\mathbb{R}^{n}}\frac{e^{-\frac{|x-y|^{2}}{4(t-\tau)}}}{(4\pi(t-\tau))^{\frac{n}{2}}}|u(y,\tau)|^{b-1}u(y,\tau)\,dy\,d\tau\\
&\quad+\int_{\frac{t}{2}}^{t}\int_{\mathbb{R}^{n}}\frac{e^{-\frac{|x-y|^{2}}{4(t-\tau)}}}{(4\pi(t-\tau))^{\frac{n}{2}}}|u(y,\tau)|^{b-1}u(y,\tau)\,dy\,d\tau\\
&=:T_{1}(u)(x,t)+T_{2}(u)(x,t).
\end{aligned} T ( u ) ( x , t ) = ∫ 0 2 t ∫ R n ( 4 π ( t − τ ) ) 2 n e − 4 ( t − τ ) ∣ x − y ∣ 2 ∣ u ( y , τ ) ∣ b − 1 u ( y , τ ) d y d τ + ∫ 2 t t ∫ R n ( 4 π ( t − τ ) ) 2 n e − 4 ( t − τ ) ∣ x − y ∣ 2 ∣ u ( y , τ ) ∣ b − 1 u ( y , τ ) d y d τ =: T 1 ( u ) ( x , t ) + T 2 ( u ) ( x , t ) . By (41) , we have
∣ T 1 ( u ) ( x , t ) ∣ ≲ ∑ k ∈ Z n t − n 2 ( 1 + ∣ k ∣ ) n + 1 ∫ 0 t 2 ∫ x − y t ∈ k + [ 0 , 1 ] n ∣ u ( y , τ ) ∣ b d y d τ ≲ ∑ k ∈ Z n 1 ( 1 + ∣ k ∣ ) n + 1 t n + 2 − 2 b ′ 2 ≲ t − 1 b − 1 ∥ u ∥ Y b n t b . (42) \begin{aligned}
|T_{1}(u)(x,t)|
&\lesssim\sum_{k\in\mathbb{Z}^{n}}\frac{t^{-\frac{n}{2}}}{(1+|k|)^{n+1}}\int_{0}^{\frac{t}{2}}\int_{\frac{x-y}{\sqrt{t}}\in k+[0,1]^{n}}|u(y,\tau)|^{b}\,dy\,d\tau\\
&\lesssim\sum_{k\in\mathbb{Z}^{n}}\frac{1}{(1+|k|)^{n+1}}t^{\frac{n+2-2b'}{2}}\\
&\lesssim t^{-\frac{1}{b-1}}\|u\|_{Y_{b}^{\sqrt{n t}}}^{b}. \tag*{(42)}
\end{aligned} ∣ T 1 ( u ) ( x , t ) ∣ ≲ k ∈ Z n ∑ ( 1 + ∣ k ∣ ) n + 1 t − 2 n ∫ 0 2 t ∫ t x − y ∈ k + [ 0 , 1 ] n ∣ u ( y , τ ) ∣ b d y d τ ≲ k ∈ Z n ∑ ( 1 + ∣ k ∣ ) n + 1 1 t 2 n + 2 − 2 b ′ ≲ t − b − 1 1 ∥ u ∥ Y b n t b . (42) On the other hand, as the heat semigroup is a contraction on L p L^p L p 1 ≤ p ≤ ∞ 1 \le p \le\infty 1 ≤ p ≤ ∞
∣ T 2 ( u ) ( x , t ) ∣ ≲ ∫ t / 2 t ∥ u ( ⋅ , τ ) ∥ L ∞ b d τ ≲ ∫ t / 2 t τ − b b − 1 d τ sup s > 0 s b b − 1 ∥ u ( ⋅ , s ) ∥ L ∞ b ≲ t − 1 b − 1 sup 0 < s < t s b b − 1 ∥ u ( ⋅ , s ) ∥ L ∞ b . (43) \begin{aligned}
|T_2(u)(x,t)| &\lesssim\int_{t/2}^{t} \|u(\cdot,\tau)\|_{L^\infty}^{b}\,d\tau\\
&\lesssim\int_{t/2}^{t} \tau^{-\frac{b}{b-1}}\,d\tau\sup_{s>0}s^{\frac{b}{b-1}}\|u(\cdot,s)\|_{L^\infty}^{b} \\
&\lesssim t^{-\frac{1}{b-1}}\sup_{0<s<t}s^{\frac{b}{b-1}}\|u(\cdot,s)\|_{L^\infty}^{b}. \tag*{(43)}
\end{aligned} ∣ T 2 ( u ) ( x , t ) ∣ ≲ ∫ t /2 t ∥ u ( ⋅ , τ ) ∥ L ∞ b d τ ≲ ∫ t /2 t τ − b − 1 b d τ s > 0 sup s b − 1 b ∥ u ( ⋅ , s ) ∥ L ∞ b ≲ t − b − 1 1 0 < s < t sup s b − 1 b ∥ u ( ⋅ , s ) ∥ L ∞ b . (43) Combining estimates (42) , (43) , and the fact that cap 2 , 1 , b ′ ( Q r ) = c r n + 2 − 2 b ′ \operatorname{cap}_{2,1,b'}(Q_r)=cr^{n+2-2b'} cap 2 , 1 , b ′ ( Q r ) = c r n + 2 − 2 b ′ r > 0 r>0 r > 0 t > 0 t>0 t > 0
t 1 b − 1 ∥ T ( u ) ( ⋅ , t ) ∥ L ∞ ≲ ∥ u ∥ Y ~ b b n t + sup 0 < s < t s b b − 1 ∥ u ( ⋅ , s ) ∥ L ∞ b ≲ ∥ u ∥ X b . t^{\frac{1}{b-1}}\|T(u)(\cdot,t)\|_{L^\infty}\lesssim\|u\|_{\widetilde{Y}_{b}^{b}\sqrt{nt}}+\sup_{0<s<t}s^{\frac{b}{b-1}}\|u(\cdot,s)\|_{L^\infty}^{b}\lesssim\|u\|_{X}^{b}. t b − 1 1 ∥ T ( u ) ( ⋅ , t ) ∥ L ∞ ≲ ∥ u ∥ Y b b n t + 0 < s < t sup s b − 1 b ∥ u ( ⋅ , s ) ∥ L ∞ b ≲ ∥ u ∥ X b . Finally, we notice that since
∣ T ( u ) ( x , t ) ∣ ≲ I 2 [ ∣ u ∣ b χ ( 0 , ∞ ) ( ⋅ ) ] ( x , t ) , ( x , t ) ∈ R + n + 1 , |T(u)(x,t)|\lesssim\mathbb{I}_2[|u|^b\chi_{(0,\infty)}(\cdot)](x,t),\qquad(x,t)\in\mathbb{R}^{n+1}_{+}, ∣ T ( u ) ( x , t ) ∣ ≲ I 2 [ ∣ u ∣ b χ ( 0 , ∞ ) ( ⋅ )] ( x , t ) , ( x , t ) ∈ R + n + 1 , it follows from Proposition 2.4 and Theorem 2.2 that
∥ T ( u ) ∥ Y b , b ′ ≲ ∥ u ∥ Y b , b ′ b ≲ ∥ u ∥ X b . \|T(u)\|_{Y_{b,b'}}\lesssim\|u\|_{Y_{b,b'}}^{b}\lesssim\|u\|_{X}^{b}. ∥ T ( u ) ∥ Y b , b ′ ≲ ∥ u ∥ Y b , b ′ b ≲ ∥ u ∥ X b . A consequence of Lemma 4.7 is the following local Lipschitz estimate for the operator T T T
Lemma 4.8. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 u 1 , u 2 ∈ X u_1,u_2\in X u 1 , u 2 ∈ X
∥ T ( u 1 ) − T ( u 2 ) ∥ X ≤ C ∥ u 1 − u 2 ∥ X ( ∥ u 1 ∥ X + ∥ u 2 ∥ X ) b − 1 . \|T(u_1)-T(u_2)\|_X\leq C\|u_1-u_2\|_X(\|u_1\|_X+\|u_2\|_X)^{b-1}. ∥ T ( u 1 ) − T ( u 2 ) ∥ X ≤ C ∥ u 1 − u 2 ∥ X ( ∥ u 1 ∥ X + ∥ u 2 ∥ X ) b − 1 . Proof. Using the inequality
∣ A ∣ b − 1 A − ∣ B ∣ b − 1 B ∣ ≤ b ∣ A − B ∣ ( ∣ A ∣ b − 1 + ∣ B ∣ b − 1 ) , ∀ A , B ∈ R , (44) |A|^{b-1}A-|B|^{b-1}B|\leq b|A-B|(|A|^{b-1}+|B|^{b-1}),\qquad\forall A,B\in\mathbb{R}, \tag*{(44)} ∣ A ∣ b − 1 A − ∣ B ∣ b − 1 B ∣ ≤ b ∣ A − B ∣ ( ∣ A ∣ b − 1 + ∣ B ∣ b − 1 ) , ∀ A , B ∈ R , (44) we have
∣ T ( u 1 ) − T ( u 2 ) ∣ ≤ b H ∗ [ ∣ u 1 − u 2 ∣ ( ∣ u 1 ∣ b − 1 + ∣ u 2 ∣ b − 1 ) χ ( 0 , ∞ ) ( ⋅ ) ] . |T(u_1)-T(u_2)|\leq bH*\left[|u_1-u_2|(|u_1|^{b-1}+|u_2|^{b-1})\chi_{(0,\infty)}(\cdot)\right]. ∣ T ( u 1 ) − T ( u 2 ) ∣ ≤ b H ∗ [ ∣ u 1 − u 2 ∣ ( ∣ u 1 ∣ b − 1 + ∣ u 2 ∣ b − 1 ) χ ( 0 , ∞ ) ( ⋅ ) ] . Then by Lemma 4.7 , we find
∥ T ( u 1 ) − T ( u 2 ) ∥ X ≲ ∥ ∣ u 1 − u 2 ∣ 1 b ( ∣ u 1 ∣ b − 1 + ∣ u 2 ∣ b − 1 ) 1 b ∥ X b ≲ ∥ u 1 − u 2 ∥ X ( ∥ u 1 ∥ X + ∥ u 2 ∥ X ) b − 1 , \|T(u_1)-T(u_2)\|_X\lesssim\left\||u_1-u_2|^{\frac{1}{b}}(|u_1|^{b-1}+|u_2|^{b-1})^{\frac{1}{b}}\right\|_X^{b}
\lesssim\|u_1-u_2\|_X(\|u_1\|_X+\|u_2\|_X)^{b-1}, ∥ T ( u 1 ) − T ( u 2 ) ∥ X ≲ ∣ u 1 − u 2 ∣ b 1 ( ∣ u 1 ∣ b − 1 + ∣ u 2 ∣ b − 1 ) b 1 X b ≲ ∥ u 1 − u 2 ∥ X ( ∥ u 1 ∥ X + ∥ u 2 ∥ X ) b − 1 , where we used Hölder’s inequality in the last inequality.
The following theorem provides our main existence result for signed initial data u 0 u_0 u 0 Z b , b ′ ( R n ) Z_{b,b'}(\mathbb{R}^n) Z b , b ′ ( R n ) b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2
Theorem 4.9. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 u 0 ∈ S ′ ( R n ) u_0\in\mathcal{S}'(\mathbb{R}^n) u 0 ∈ S ′ ( R n ) ε = ε ( n , b ) > 0 \varepsilon=\varepsilon(n,b)>0 ε = ε ( n , b ) > 0 u 0 ∈ Z b , b ′ ( R n ) u_0\in Z_{b,b'}(\mathbb{R}^n) u 0 ∈ Z b , b ′ ( R n )
∥ u 0 ∥ Z b , b ′ ( R n ) = ∥ e t Δ u 0 ∥ Y b , b ′ ≤ ε , \|u_0\|_{Z_{b,b'}(\mathbb{R}^n)}=\|e^{t\Delta}u_0\|_{Y_{b,b'}}\leq\varepsilon, ∥ u 0 ∥ Z b , b ′ ( R n ) = ∥ e t Δ u 0 ∥ Y b , b ′ ≤ ε , then there exists a unique small global solution u ∈ X u \in X u ∈ X (24) , which satisfies the estimate
∣ u ( x , t ) ∣ ≤ ∣ e t Δ u 0 ( x ) ∣ + C I 2 ( ∣ e τ Δ u 0 ∣ b ) ( x , t ) (45) |u(x,t)| \leq|e^{t\Delta}u_0(x)| + C\mathbb{I}_2(|e^{\tau\Delta}u_0|^b)(x,t) \tag*{(45)} ∣ u ( x , t ) ∣ ≤ ∣ e t Δ u 0 ( x ) ∣ + C I 2 ( ∣ e τ Δ u 0 ∣ b ) ( x , t ) (45) for a.e. ( x , t ) ∈ R n × ( 0 , ∞ ) (x,t) \in\mathbb{R}^n \times(0,\infty) ( x , t ) ∈ R n × ( 0 , ∞ )
Proof. For u 0 ∈ Z b , b ′ ( R n ) u_0 \in Z_{b,b'}(\mathbb{R}^n) u 0 ∈ Z b , b ′ ( R n ) v ∈ X v \in X v ∈ X
S ( v ) = e t Δ u 0 + ∫ 0 t e ( t − τ ) Δ ( ∣ v ( ⋅ , τ ) ∣ b − 1 v ( ⋅ , τ ) ) d τ = e t Δ u 0 + T ( v ) . S(v) = e^{t\Delta}u_0 + \int_0^t e^{(t-\tau)\Delta}\left(|v(\cdot,\tau)|^{b-1}v(\cdot,\tau)\right)\,d\tau
= e^{t\Delta}u_0 + T(v). S ( v ) = e t Δ u 0 + ∫ 0 t e ( t − τ ) Δ ( ∣ v ( ⋅ , τ ) ∣ b − 1 v ( ⋅ , τ ) ) d τ = e t Δ u 0 + T ( v ) . Since b > 1 b>1 b > 1 4.7 , and 4.8 , there exist ε 1 = ε 1 ( n , b ) > 0 \varepsilon_1=\varepsilon_1(n,b)>0 ε 1 = ε 1 ( n , b ) > 0 δ = δ ( n , b ) > 0 \delta=\delta(n,b)>0 δ = δ ( n , b ) > 0 ε ∈ ( 0 , ε 1 ] \varepsilon\in(0,\varepsilon_1] ε ∈ ( 0 , ε 1 ]
S : B δ X ‾ → B δ X ‾ S:\overline{B_\delta^X}\to\overline{B_\delta^X} S : B δ X → B δ X is a contraction, where B δ X ‾ = { v ∈ X : ∥ v ∥ X ≤ δ } \overline{B_\delta^X}=\{v\in X:\|v\|_X\leq\delta\} B δ X = { v ∈ X : ∥ v ∥ X ≤ δ } u ∈ B δ X ‾ u\in\overline{B_\delta^X} u ∈ B δ X u = S ( u ) u=S(u) u = S ( u )
u = lim N → ∞ u N in X , u=\lim_{N\to\infty}u_N\quad\text{in }X, u = N → ∞ lim u N in X , where the sequence of functions { u N } N ∈ N \{u_N\}_{N\in\mathbb{N}} { u N } N ∈ N u 1 ( t ) = e t Δ u 0 u_1(t)=e^{t\Delta}u_0 u 1 ( t ) = e t Δ u 0
u N + 1 ( t ) = e t Δ u 0 + H ∗ ( ∣ u N ∣ b − 1 u N χ ( 0 , ∞ ) ( ⋅ ) ) , ∀ N ≥ 1. (46) u_{N+1}(t)=e^{t\Delta}u_0+H*(|u_N|^{b-1}u_N\chi_{(0,\infty)}(\cdot)),\qquad\forall N\geq1. \tag*{(46)} u N + 1 ( t ) = e t Δ u 0 + H ∗ ( ∣ u N ∣ b − 1 u N χ ( 0 , ∞ ) ( ⋅ )) , ∀ N ≥ 1. (46) It remains to show the pointwise estimate (45) . To this end, let U ( t ) = e t Δ u 0 = e t Δ u 0 χ ( 0 , ∞ ) ( t ) U(t)=e^{t\Delta}u_0=e^{t\Delta}u_0\chi_{(0,\infty)}(t) U ( t ) = e t Δ u 0 = e t Δ u 0 χ ( 0 , ∞ ) ( t ) Theorem 2.2 , we have
∫ E ∣ U ∣ b d x d t ≤ c 1 ε b Cap I 2 , b ′ ( E ) \int_E |U|^b\,dx\,dt\leq c_1\varepsilon^b\operatorname{Cap}_{\mathbb{I}_2,b'}(E) ∫ E ∣ U ∣ b d x d t ≤ c 1 ε b Cap I 2 , b ′ ( E ) for all compact sets E ⊂ R n + 1 E\subset\mathbb{R}^{n+1} E ⊂ R n + 1 Proposition 2.4 ,
I 2 ( I 2 ( ∣ U ∣ b ) ) b ≤ c 2 ε b ( b − 1 ) I 2 ( ∣ U ∣ b ) < + ∞ a.e. (47) \mathbb{I}_2\left(\mathbb{I}_2(|U|^b)\right)^b\leq c_2\varepsilon^{b(b-1)}\mathbb{I}_2(|U|^b)<+\infty\quad\text{a.e.} \tag*{(47)} I 2 ( I 2 ( ∣ U ∣ b ) ) b ≤ c 2 ε b ( b − 1 ) I 2 ( ∣ U ∣ b ) < + ∞ a.e. (47) We next claim that there exists a positive constant ε 2 ≤ ε 1 \varepsilon_2\leq\varepsilon_1 ε 2 ≤ ε 1 ε ∈ ( 0 , ε 2 ] \varepsilon\in(0,\varepsilon_2] ε ∈ ( 0 , ε 2 ]
∣ u N ∣ ≤ ∣ U ∣ + C I 2 ( ∣ U ∣ b ) , ∀ N ≥ 1. (48) |u_N|\leq|U|+C\mathbb{I}_2(|U|^b),\qquad\forall N\geq1. \tag*{(48)} ∣ u N ∣ ≤ ∣ U ∣ + C I 2 ( ∣ U ∣ b ) , ∀ N ≥ 1. (48) Indeed, ∣ u 1 ∣ ≤ ∣ U ∣ |u_1|\leq|U| ∣ u 1 ∣ ≤ ∣ U ∣ N ≥ 1 N\geq1 N ≥ 1
∣ u N ∣ ≤ ∣ U ∣ + C N I 2 ( ∣ U ∣ b ) . |u_N|\leq|U|+C_N\mathbb{I}_2(|U|^b). ∣ u N ∣ ≤ ∣ U ∣ + C N I 2 ( ∣ U ∣ b ) . Then it follows from (47) that
∣ u N + 1 ∣ ≤ ∣ U ∣ + c 3 I 2 [ ( C N ) b I 2 ( ∣ U ∣ b ) b ] + c 3 I 2 [ ∣ U ∣ b ] ≤ ∣ U ∣ + c 3 I 2 [ ∣ U ∣ b ] { ( C N ) b c 2 ε b ( b − 1 ) + 1 } , \begin{aligned}
|u_{N+1}|\leq|U|+c_3\mathbb{I}_2\left[(C_N)^b\mathbb{I}_2(|U|^b)^b\right]+c_3\mathbb{I}_2[|U|^b] \\
&\leq|U|+c_3\mathbb{I}_2[|U|^b]\left\{(C_N)^b c_2\varepsilon^{b(b-1)}+1\right\},
\end{aligned} ∣ u N + 1 ∣ ≤ ∣ U ∣ + c 3 I 2 [ ( C N ) b I 2 ( ∣ U ∣ b ) b ] + c 3 I 2 [ ∣ U ∣ b ] ≤ ∣ U ∣ + c 3 I 2 [ ∣ U ∣ b ] { ( C N ) b c 2 ε b ( b − 1 ) + 1 } , where c 3 = c 3 ( n , b ) c_3=c_3(n,b) c 3 = c 3 ( n , b )
∣ u N ∣ ≤ ∣ U ∣ + C N I 2 [ ∣ U ∣ b ] , ∀ N ≥ 1 , |u_N|\leq|U|+C_N\mathbb{I}_2[|U|^b],\qquad\forall N\geq1, ∣ u N ∣ ≤ ∣ U ∣ + C N I 2 [ ∣ U ∣ b ] , ∀ N ≥ 1 , with
C 1 = 0 and C N + 1 = c 3 { ( C N ) b c 2 ε b ( b − 1 ) + 1 } for N ≥ 1. C_1=0 \quad\text{and}\quad C_{N+1}=c_3\left\{(C_N)^b c_2\varepsilon^{b(b-1)}+1\right\}\quad\text{for }N\geq1. C 1 = 0 and C N + 1 = c 3 { ( C N ) b c 2 ε b ( b − 1 ) + 1 } for N ≥ 1. By induction, it is then easy to see that C N ≤ c 3 b ′ C_N\leq c_3b' C N ≤ c 3 b ′ N ≥ 1 N\geq1 N ≥ 1 ε ≤ ε 2 \varepsilon\leq\varepsilon_2 ε ≤ ε 2 ε 2 ≤ ε 1 \varepsilon_2\leq\varepsilon_1 ε 2 ≤ ε 1
c 2 ε 2 b ( b − 1 ) ≤ ( c 3 ) − b ( b − 1 ) b − 1 b − b . c_2\varepsilon_2^{b(b-1)}\leq(c_3)^{-b}(b-1)^{b-1}b^{-b}. c 2 ε 2 b ( b − 1 ) ≤ ( c 3 ) − b ( b − 1 ) b − 1 b − b . This proves the claim (48) with C = c 3 b ′ C=c_3b' C = c 3 b ′
To continue, we will now show that there exists c = c ( n ) > 0 c=c(n)>0 c = c ( n ) > 0
∣ u N + 1 − u N ∣ ≤ c 2 − N I 2 ( ∣ U ∣ b ) , ∀ N ≥ 1. (49) \lvert u_{N+1}-u_N\rvert\leq c2^{-N}\mathbb{I}_2(\lvert U\rvert^b),\quad\forall N\geq1. \tag*{(49)} ∣ u N + 1 − u N ∣ ≤ c 2 − N I 2 (∣ U ∣ b ) , ∀ N ≥ 1. (49) To see this, note that
∣ u 2 − u 1 ∣ ≤ a 1 I 2 ( ∣ U ∣ b ) , \lvert u_2-u_1\rvert\leq a_1\mathbb{I}_2(\lvert U\rvert^b), ∣ u 2 − u 1 ∣ ≤ a 1 I 2 (∣ U ∣ b ) , and by induction, suppose that
∣ u N − u N − 1 ∣ ≤ a N − 1 I 2 ( ∣ U ∣ b ) (50) \lvert u_N-u_{N-1}\rvert\leq a_{N-1}\mathbb{I}_2(\lvert U\rvert^b) \tag*{(50)} ∣ u N − u N − 1 ∣ ≤ a N − 1 I 2 (∣ U ∣ b ) (50) for some N ≥ 2 N\geq2 N ≥ 2 (46) , (50) , and inequality (44) , we bound
∣ u N + 1 − u N ∣ ≤ C I 2 [ ∣ u N ∣ b − 1 u N − ∣ u N − 1 ∣ b − 1 u N − 1 ] ≤ C I 2 [ ∣ u N − u N − 1 ∣ ( ∣ u N ∣ b − 1 + ∣ u N − 1 ∣ b − 1 ) ] ≤ C a N − 1 I 2 [ I 2 ( ∣ U ∣ b ) ( ∣ u N ∣ b − 1 + ∣ u N − 1 ∣ b − 1 ) ] . \begin{aligned}
\lvert u_{N+1}-u_N\rvert
&\leq C\mathbb{I}_2\left[\lvert u_N\rvert^{b-1}u_N-\lvert u_{N-1}\rvert^{b-1}u_{N-1}\right]\\
&\leq C\mathbb{I}_2\left[\lvert u_N-u_{N-1}\rvert\left(\lvert u_N\rvert^{b-1}+\lvert u_{N-1}\rvert^{b-1}\right)\right]\\
&\leq Ca_{N-1}\mathbb{I}_2\left[\mathbb{I}_2(\lvert U\rvert^b)\left(\lvert u_N\rvert^{b-1}+\lvert u_{N-1}\rvert^{b-1}\right)\right].
\end{aligned} ∣ u N + 1 − u N ∣ ≤ C I 2 [ ∣ u N ∣ b − 1 u N − ∣ u N − 1 ∣ b − 1 u N − 1 ] ≤ C I 2 [ ∣ u N − u N − 1 ∣ ( ∣ u N ∣ b − 1 + ∣ u N − 1 ∣ b − 1 ) ] ≤ C a N − 1 I 2 [ I 2 (∣ U ∣ b ) ( ∣ u N ∣ b − 1 + ∣ u N − 1 ∣ b − 1 ) ] . where C = C ( n , b ) C=C(n,b) C = C ( n , b ) (48) , we obtain
∣ u N + 1 − u N ∣ ≤ C a N − 1 I 2 [ I 2 ( ∣ U ∣ b ) { I 2 ( ∣ U ∣ b ) b − 1 + ∣ U ∣ b − 1 } ] ≤ C a N − 1 I 2 [ I 2 ( ∣ U ∣ b ) b ] + C a N − 1 I 2 [ I 2 ( ∣ U ∣ b ) ∣ U ∣ b − 1 ] . \begin{aligned}
\lvert u_{N+1}-u_N\rvert
&\leq Ca_{N-1}\mathbb{I}_2\left[\mathbb{I}_2(\lvert U\rvert^b)\left\{\mathbb{I}_2(\lvert U\rvert^b)^{b-1}+\lvert U\rvert^{b-1}\right\}\right]\\
&\leq Ca_{N-1}\mathbb{I}_2\left[\mathbb{I}_2(\lvert U\rvert^b)^b\right]+Ca_{N-1}\mathbb{I}_2\left[\mathbb{I}_2(\lvert U\rvert^b)\lvert U\rvert^{b-1}\right].
\end{aligned} ∣ u N + 1 − u N ∣ ≤ C a N − 1 I 2 [ I 2 (∣ U ∣ b ) { I 2 (∣ U ∣ b ) b − 1 + ∣ U ∣ b − 1 } ] ≤ C a N − 1 I 2 [ I 2 (∣ U ∣ b ) b ] + C a N − 1 I 2 [ I 2 (∣ U ∣ b ) ∣ U ∣ b − 1 ] . Note that, by Young’s inequality,
( I 2 ∣ U ∣ b ) ∣ U ∣ b − 1 ≤ c ( β ) ( I 2 ∣ U ∣ b ) b + β ∣ U ∣ b , ∀ β > 0 , \left(\mathbb{I}_2\lvert U\rvert^b\right)\lvert U\rvert^{b-1}\leq c(\beta)\left(\mathbb{I}_2\lvert U\rvert^b\right)^b+\beta\lvert U\rvert^b,\quad\forall\beta>0, ( I 2 ∣ U ∣ b ) ∣ U ∣ b − 1 ≤ c ( β ) ( I 2 ∣ U ∣ b ) b + β ∣ U ∣ b , ∀ β > 0 , which then yields
∣ u N + 1 − u N ∣ ≤ { C a N − 1 ( 1 + c ( β ) ) } I 2 [ I 2 ( ∣ U ∣ b ) b ] + C a N − 1 β I 2 [ ∣ U ∣ b ] ≤ { C a N − 1 ( 1 + c ( β ) ) } c 2 ε b ( b − 1 ) I 2 [ ∣ U ∣ b ] + C a N − 1 β I 2 [ ∣ U ∣ b ] , \begin{aligned}
\lvert u_{N+1}-u_N\rvert
&\leq\left\{Ca_{N-1}(1+c(\beta))\right\}\mathbb{I}_2\left[\mathbb{I}_2(\lvert U\rvert^b)^b\right]+Ca_{N-1}\beta\mathbb{I}_2[\lvert U\rvert^b]\\
&\leq\left\{Ca_{N-1}(1+c(\beta))\right\}c_2\varepsilon^{b(b-1)}\mathbb{I}_2[\lvert U\rvert^b]+Ca_{N-1}\beta\mathbb{I}_2[\lvert U\rvert^b],
\end{aligned} ∣ u N + 1 − u N ∣ ≤ { C a N − 1 ( 1 + c ( β )) } I 2 [ I 2 (∣ U ∣ b ) b ] + C a N − 1 β I 2 [∣ U ∣ b ] ≤ { C a N − 1 ( 1 + c ( β )) } c 2 ε b ( b − 1 ) I 2 [∣ U ∣ b ] + C a N − 1 β I 2 [∣ U ∣ b ] , where we used (47) in the last bound. Hence,
∣ u N + 1 − u N ∣ ≤ a N I 2 [ ∣ U ∣ b ] , \lvert u_{N+1}-u_N\rvert\leq a_N\mathbb{I}_2[\lvert U\rvert^b], ∣ u N + 1 − u N ∣ ≤ a N I 2 [∣ U ∣ b ] , with a N ≤ 1 2 a N − 1 a_N\leq\frac{1}{2}a_{N-1} a N ≤ 2 1 a N − 1 β > 0 \beta>0 β > 0 C β ≤ 1 / 4 C\beta\leq1/4 C β ≤ 1/4 ε ∈ ( 0 , ε 3 ] \varepsilon\in(0,\varepsilon_3] ε ∈ ( 0 , ε 3 ] ε 3 ≤ ε 2 \varepsilon_3\leq\varepsilon_2 ε 3 ≤ ε 2
{ C ( 1 + c ( β ) ) } c 2 ε 3 b ( b − 1 ) ≤ 1 / 4. \{C(1+c(\beta))\}c_2\varepsilon_3^{b(b-1)}\leq1/4. { C ( 1 + c ( β ))} c 2 ε 3 b ( b − 1 ) ≤ 1/4. This yields (49) with c = 2 a 1 c=2a_1 c = 2 a 1 (49) , we have the pointwise limit
u = lim N → ∞ u N = u 1 + ∑ N = 1 ∞ ( u N + 1 − u N ) , (51) u=\lim_{N\to\infty}u_N=u_1+\sum_{N=1}^{\infty}(u_{N+1}-u_N), \tag*{(51)} u = N → ∞ lim u N = u 1 + N = 1 ∑ ∞ ( u N + 1 − u N ) , (51) where the series converges absolutely pointwise a.e. Finally, letting N → ∞ N\to\infty N → ∞ (48) , we arrive at the pointwise estimate (45) .
Our next goal is to show the regularity/persistency of solutions obtained in Theorem 4.9 when the initial datum u 0 u_0 u 0 Z b , b ′ ( R n ) ∩ L a ( R n ) Z_{b,b'}(\mathbb{R}^n)\cap L^a(\mathbb{R}^n) Z b , b ′ ( R n ) ∩ L a ( R n ) a > 1 a>1 a > 1
Lemma 4.10. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 a > 1 a>1 a > 1 u , v ∈ X ∩ L t ∞ L x a u,v\in X\cap L^\infty_tL^a_x u , v ∈ X ∩ L t ∞ L x a
∥ T ( u ) − T ( v ) ∥ L t ∞ L x a ≤ C ∥ u − v ∥ X b − 1 b ∥ ∣ u ∣ + ∣ v ∣ ∥ X ( b − 1 ) 2 b ∥ ∣ u ∣ + ∣ v ∣ ∥ L t ∞ L x a . \|T(u)-T(v)\|_{L^\infty_tL^a_x}\le C\|u-v\|_X^{\frac{b-1}{b}}\||u|+|v|\|_X^{\frac{(b-1)^2}{b}}\||u|+|v|\|_{L^\infty_tL^a_x}. ∥ T ( u ) − T ( v ) ∥ L t ∞ L x a ≤ C ∥ u − v ∥ X b b − 1 ∥∣ u ∣ + ∣ v ∣ ∥ X b ( b − 1 ) 2 ∥∣ u ∣ + ∣ v ∣ ∥ L t ∞ L x a . Proof. By (44) , we have
∣ T ( u ) ( x , t ) − T ( v ) ( x , t ) ∣ ≤ C ∫ 0 t e ( t − τ ) Δ h ( u , v ) ( x ) d τ , (52) |T(u)(x,t)-T(v)(x,t)|\le C\int_0^t e^{(t-\tau)\Delta}h(u,v)(x)\,d\tau, \tag*{(52)} ∣ T ( u ) ( x , t ) − T ( v ) ( x , t ) ∣ ≤ C ∫ 0 t e ( t − τ ) Δ h ( u , v ) ( x ) d τ , (52) where h ( u , v ) = ∣ u − v ∣ ( ∣ u ∣ b − 1 + ∣ v ∣ b − 1 ) h(u,v)=|u-v|(|u|^{b-1}+|v|^{b-1}) h ( u , v ) = ∣ u − v ∣ ( ∣ u ∣ b − 1 + ∣ v ∣ b − 1 )
Suppose first that 1 + 2 n < b ≤ a 1+\frac{2}{n}<b\le a 1 + n 2 < b ≤ a (52) and by Lemma 4.7 , we find
∥ [ T ( u ) − T ( v ) ] ( ⋅ , t ) ∥ L x a ≤ ∥ [ T ( u ) − T ( v ) ] ( ⋅ , t ) ∥ L x ∞ b − 1 b ∥ [ T ( u ) − T ( v ) ] ( ⋅ , t ) ∥ L x a b 1 b \|[T(u)-T(v)](\cdot,t)\|_{L^a_x}
\le\|[T(u)-T(v)](\cdot,t)\|_{L^\infty_x}^{\frac{b-1}{b}}\|[T(u)-T(v)](\cdot,t)\|_{L^{\frac{a}{b}}_x}^{\frac{1}{b}} ∥ [ T ( u ) − T ( v )] ( ⋅ , t ) ∥ L x a ≤ ∥ [ T ( u ) − T ( v )] ( ⋅ , t ) ∥ L x ∞ b b − 1 ∥ [ T ( u ) − T ( v )] ( ⋅ , t ) ∥ L x b a b 1 ≲ t − 1 b ∥ h ( u , v ) ∥ X 1 b ( ∫ 0 t ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x a b a b d τ ) 1 b \lesssim t^{-\frac{1}{b}}\|h(u,v)\|_X^{\frac{1}{b}}\left(\int_0^t\|h(u,v)(\cdot,\tau)\|_{L^{\frac{a}{b}}_x}^{\frac{a}{b}}\,d\tau\right)^{\frac{1}{b}} ≲ t − b 1 ∥ h ( u , v ) ∥ X b 1 ( ∫ 0 t ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x b a b a d τ ) b 1 ≲ ∥ u − v ∥ X b − 1 b ∥ ∣ u ∣ + ∣ v ∣ ∥ X ( b − 1 ) 2 b sup τ ∈ [ 0 , ∞ ) ∥ u ( ⋅ , τ ) + v ( ⋅ , τ ) ∥ L x a . \lesssim\|u-v\|_X^{\frac{b-1}{b}}\||u|+|v|\|_X^{\frac{(b-1)^2}{b}}\sup_{\tau\in[0,\infty)}\|u(\cdot,\tau)+v(\cdot,\tau)\|_{L^a_x}. ≲ ∥ u − v ∥ X b b − 1 ∥∣ u ∣ + ∣ v ∣ ∥ X b ( b − 1 ) 2 τ ∈ [ 0 , ∞ ) sup ∥ u ( ⋅ , τ ) + v ( ⋅ , τ ) ∥ L x a . Thus we obtain the lemma in the case b ≤ a b\le a b ≤ a
Suppose now that b > a > 1 b>a>1 b > a > 1
∥ [ T ( u ) − T ( v ) ] ( ⋅ , t ) ∥ L x a ≤ ∥ [ T ( u ) − T ( v ) ] ( ⋅ , t ) ∥ L x a a − 1 a − 1 a ∥ [ T ( u ) − T ( v ) ] ( ⋅ , t ) ∥ L x 1 1 a \|[T(u)-T(v)](\cdot,t)\|_{L^a_x}
\le\|[T(u)-T(v)](\cdot,t)\|_{L_x^{\frac{a}{a-1}}}^{\frac{a-1}{a}}\|[T(u)-T(v)](\cdot,t)\|_{L^1_x}^{\frac{1}{a}} ∥ [ T ( u ) − T ( v )] ( ⋅ , t ) ∥ L x a ≤ ∥ [ T ( u ) − T ( v )] ( ⋅ , t ) ∥ L x a − 1 a a a − 1 ∥ [ T ( u ) − T ( v )] ( ⋅ , t ) ∥ L x 1 a 1 ≲ t − 1 b − 1 a − 1 a ∥ h ( u , v ) ∥ X 1 b b ( a − 1 ) a ( ∫ 0 t ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x 1 d τ ) 1 a . \lesssim t^{-\frac{1}{b-1}\frac{a-1}{a}}\|h(u,v)\|_X^{\frac{1}{b}\frac{b(a-1)}{a}}\left(\int_0^t\|h(u,v)(\cdot,\tau)\|_{L^1_x}\,d\tau\right)^{\frac{1}{a}}. ≲ t − b − 1 1 a a − 1 ∥ h ( u , v ) ∥ X b 1 a b ( a − 1 ) ( ∫ 0 t ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x 1 d τ ) a 1 . Also, note that
∥ h ( u , v ) ( ⋅ , τ ) ∥ L x 1 ≤ ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x b b − a 1 b ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x a 1 b \|h(u,v)(\cdot,\tau)\|_{L^1_x}\le\|h(u,v)(\cdot,\tau)\|_{L_x^{\frac{b}{b-a}}}^{\frac{1}{b}}\|h(u,v)(\cdot,\tau)\|_{L^a_x}^{\frac{1}{b}} ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x 1 ≤ ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x b − a b b 1 ∥ h ( u , v ) ( ⋅ , τ ) ∥ L x a b 1 ≲ ∥ h ( u , v ) ∥ X 1 b τ − b − a b − 1 1 b sup τ ∈ [ 0 , ∞ ) ∥ u ( ⋅ , τ ) + v ( ⋅ , τ ) ∥ L x a a . \lesssim\|h(u,v)\|_X^{\frac{1}{b}}\tau^{-\frac{b-a}{b-1}\frac{1}{b}}\sup_{\tau\in[0,\infty)}\|u(\cdot,\tau)+v(\cdot,\tau)\|_{L^a_x}^{a}. ≲ ∥ h ( u , v ) ∥ X b 1 τ − b − 1 b − a b 1 τ ∈ [ 0 , ∞ ) sup ∥ u ( ⋅ , τ ) + v ( ⋅ , τ ) ∥ L x a a . Thus, since a > 1 a>1 a > 1
∥ [ T ( u ) − T ( v ) ] ( ⋅ , t ) ∥ L x a ≲ t − 1 b − 1 − a − 1 a ∥ h ( u , v ) 1 b ∥ X b − 1 ( ∫ 0 t τ − b − a b − 1 d τ ) 1 a ∥ ∣ u ∣ + ∣ v ∣ ∥ L t ∞ L x a ≲ ∥ u − v ∥ X b − 1 b ∥ ∣ u ∣ + ∣ v ∣ ∥ X ( b − 1 ) 2 b ∥ ∣ u ∣ + ∣ v ∣ ∥ L t ∞ L x a . \begin{aligned}
\|[T(u)-T(v)](\cdot,t)\|_{L_x^a}
&\lesssim t^{-\frac{1}{b-1}-\frac{a-1}{a}}\|h(u,v)^{\frac{1}{b}}\|_X^{b-1}\left(\int_0^t\tau^{-\frac{b-a}{b-1}}\ d\tau\right)^{\frac{1}{a}}\||u|+|v|\|_{L_t^\infty L_x^a}\\
&\lesssim\|u-v\|_X^{\frac{b-1}{b}}\||u|+|v|\|_X^{\frac{(b-1)^2}{b}}\||u|+|v|\|_{L_t^\infty L_x^a}.
\end{aligned} ∥ [ T ( u ) − T ( v )] ( ⋅ , t ) ∥ L x a ≲ t − b − 1 1 − a a − 1 ∥ h ( u , v ) b 1 ∥ X b − 1 ( ∫ 0 t τ − b − 1 b − a d τ ) a 1 ∥∣ u ∣ + ∣ v ∣ ∥ L t ∞ L x a ≲ ∥ u − v ∥ X b b − 1 ∥∣ u ∣ + ∣ v ∣ ∥ X b ( b − 1 ) 2 ∥∣ u ∣ + ∣ v ∣ ∥ L t ∞ L x a . This proves the lemma in the case b > a b>a b > a
We next prove a persistency result in L a ( R n ) L^a(\mathbb{R}^n) L a ( R n ) 1 < a < ∞ 1<a<\infty 1 < a < ∞
Theorem 4.11. Let b > 1 + 2 n b>1+\frac{2}{n} b > 1 + n 2 a > 1 a>1 a > 1 u 0 ∈ Z b , b ′ ( R n ) ∩ L a ( R n ) u_0\in Z_{b,b'}(\mathbb{R}^n)\cap L^a(\mathbb{R}^n) u 0 ∈ Z b , b ′ ( R n ) ∩ L a ( R n ) u u u Theorem 4.9 also belongs to the space L ∞ ( ( 0 , ∞ ) , L a ( R n ) ) ∩ C ( ( 0 , ∞ ) , L a ( R n ) ) L^\infty((0,\infty),L^a(\mathbb{R}^n))\cap C((0,\infty),L^a(\mathbb{R}^n)) L ∞ (( 0 , ∞ ) , L a ( R n )) ∩ C (( 0 , ∞ ) , L a ( R n )) a ≥ n ( b − 1 ) 2 a\ge\frac{n(b-1)}{2} a ≥ 2 n ( b − 1 ) u ∈ C ( [ 0 , ∞ ) , L a ( R n ) ) u\in C([0,\infty),L^a(\mathbb{R}^n)) u ∈ C ([ 0 , ∞ ) , L a ( R n ))
Proof. Let S S S T T T ε = ε ( n , b ) \varepsilon=\varepsilon(n,b) ε = ε ( n , b ) δ = δ ( n , b ) \delta=\delta(n,b) δ = δ ( n , b ) { u N } N ∈ N \{u_N\}_{N\in\mathbb{N}} { u N } N ∈ N Theorem 4.9 . Recall that u 1 = e t Δ u 0 u_1=e^{t\Delta}u_0 u 1 = e t Δ u 0 S : B δ X ‾ → B δ X ‾ S:\overline{B_\delta^X}\to\overline{B_\delta^X} S : B δ X → B δ X κ = κ ( n , b ) ∈ ( 0 , 1 ) \kappa=\kappa(n,b)\in(0,1) κ = κ ( n , b ) ∈ ( 0 , 1 )
∥ T ( v 1 ) − T ( v 2 ) ∥ X = ∥ S ( v 1 ) − S ( v 2 ) ∥ X ≤ κ ∥ v 1 − v 2 ∥ X \|T(v_1)-T(v_2)\|_X=\|S(v_1)-S(v_2)\|_X\le\kappa\|v_1-v_2\|_X ∥ T ( v 1 ) − T ( v 2 ) ∥ X = ∥ S ( v 1 ) − S ( v 2 ) ∥ X ≤ κ ∥ v 1 − v 2 ∥ X for all v 1 , v 2 ∈ B δ X ‾ v_1,v_2\in\overline{B_\delta^X} v 1 , v 2 ∈ B δ X
∥ u N + 1 − u N ∥ X ≤ C κ N , ∀ N ∈ N . \|u_{N+1}-u_N\|_X\le C\kappa^N,\qquad\forall N\in\mathbb{N}. ∥ u N + 1 − u N ∥ X ≤ C κ N , ∀ N ∈ N . On the other hand, it can be shown by Lemma 4.10 and induction that { u N } N ∈ N ⊂ L t ∞ L x a \{u_N\}_{N\in\mathbb{N}}\subset L_t^\infty L_x^a { u N } N ∈ N ⊂ L t ∞ L x a Lemma 4.10 , we have, for any N ≥ 2 N\ge2 N ≥ 2
∥ u N + 1 − u N ∥ L t ∞ L x a = ∥ T ( u N ) − T ( u N − 1 ) ∥ L t ∞ L x a ≤ C κ N ( b − 1 ) b ∥ ∣ u N ∣ + ∣ u N − 1 ∣ ∥ L t ∞ L x a ≤ C κ N ( b − 1 ) b max 1 ≤ j ≤ N ∥ u j ∥ L t ∞ L x a . (53) \begin{aligned}
\|u_{N+1}-u_N\|_{L_t^\infty L_x^a}
&=\|T(u_N)-T(u_{N-1})\|_{L_t^\infty L_x^a}\\
&\le C\kappa^{\frac{N(b-1)}{b}}\||u_N|+|u_{N-1}|\|_{L_t^\infty L_x^a}\\
&\le C\kappa^{\frac{N(b-1)}{b}}\max_{1\le j\le N}\|u_j\|_{L_t^\infty L_x^a}. \tag*{(53)}
\end{aligned} ∥ u N + 1 − u N ∥ L t ∞ L x a = ∥ T ( u N ) − T ( u N − 1 ) ∥ L t ∞ L x a ≤ C κ b N ( b − 1 ) ∥∣ u N ∣ + ∣ u N − 1 ∣ ∥ L t ∞ L x a ≤ C κ b N ( b − 1 ) 1 ≤ j ≤ N max ∥ u j ∥ L t ∞ L x a . (53) where we used that { u N } N ∈ N ⊂ B δ X ‾ \{u_N\}_{N\in\mathbb{N}}\subset\overline{B_\delta^X} { u N } N ∈ N ⊂ B δ X Lemma 4.10 with u = u 1 u=u_1 u = u 1 v = 0 v=0 v = 0 (53) also holds in the case N = 1 N=1 N = 1 N ≥ 1 N\ge1 N ≥ 1
B N = max 1 ≤ j ≤ N ∥ u j ∥ L t ∞ L x a . B_N=\max_{1\le j\le N}\|u_j\|_{L_t^\infty L_x^a}. B N = 1 ≤ j ≤ N max ∥ u j ∥ L t ∞ L x a . Then as u N + 1 = ( u N + 1 − u N ) + u N u_{N+1}=(u_{N+1}-u_N)+u_N u N + 1 = ( u N + 1 − u N ) + u N
∥ u N + 1 ∥ L t ∞ L x a ≤ C κ N ( b − 1 ) b B N + B N = B N ( 1 + C κ N ( b − 1 ) b ) , \|u_{N+1}\|_{L_t^\infty L_x^a}\le C\kappa^{\frac{N(b-1)}{b}}B_N+B_N=B_N\left(1+C\kappa^{\frac{N(b-1)}{b}}\right), ∥ u N + 1 ∥ L t ∞ L x a ≤ C κ b N ( b − 1 ) B N + B N = B N ( 1 + C κ b N ( b − 1 ) ) , which yields that
B N + 1 ≤ B N ( 1 + C κ N ( b − 1 ) b ) , N ≥ 1. B_{N+1}\le B_N\left(1+C\kappa^{\frac{N(b-1)}{b}}\right),\qquad N\ge1. B N + 1 ≤ B N ( 1 + C κ b N ( b − 1 ) ) , N ≥ 1. Thus,
B N ≤ B N − 1 ( 1 + C κ N ( b − 1 ) b ) ≤ B N − 2 ( 1 + C κ N ( b − 1 ) b ) ( 1 + C κ ( N − 1 ) ( b − 1 ) b ) ≤ B 1 ∏ j = 1 ∞ ( 1 + C κ j ( b − 1 ) b ) ≤ C ∥ u 0 ∥ L a . \begin{aligned}
B_N &\le B_{N-1}\left(1+C\kappa^{\frac{N(b-1)}{b}}\right)\\
&\le B_{N-2}\left(1+C\kappa^{\frac{N(b-1)}{b}}\right)\left(1+C\kappa^{\frac{(N-1)(b-1)}{b}}\right)\\
&\le B_1\prod_{j=1}^{\infty}\left(1+C\kappa^{\frac{j(b-1)}{b}}\right)\\
&\le C\|u_0\|_{L^a}.
\end{aligned} B N ≤ B N − 1 ( 1 + C κ b N ( b − 1 ) ) ≤ B N − 2 ( 1 + C κ b N ( b − 1 ) ) ( 1 + C κ b ( N − 1 ) ( b − 1 ) ) ≤ B 1 j = 1 ∏ ∞ ( 1 + C κ b j ( b − 1 ) ) ≤ C ∥ u 0 ∥ L a . By (53) , this yields
∥ u N + 1 − u N ∥ L t ∞ L x a ≤ C κ N ( b − 1 ) b ∥ u 0 ∥ L a , ∀ N ≥ 1 , \|u_{N+1}-u_N\|_{L_t^\infty L_x^a}\le C\kappa^{\frac{N(b-1)}{b}}\|u_0\|_{L^a},\qquad\forall N\ge1, ∥ u N + 1 − u N ∥ L t ∞ L x a ≤ C κ b N ( b − 1 ) ∥ u 0 ∥ L a , ∀ N ≥ 1 , which implies that u N u_N u N u u u L t ∞ L x a L_t^\infty L_x^a L t ∞ L x a (51) , Fatou’s lemma and the above inequality, we arrive at
∥ u ∥ L t ∞ L x a ≤ C ∥ u 0 ∥ L a . \|u\|_{L_t^\infty L_x^a}\le C\|u_0\|_{L^a}. ∥ u ∥ L t ∞ L x a ≤ C ∥ u 0 ∥ L a . To show that u ∈ C ( ( 0 , ∞ ) , L a ( R n ) ) u\in C((0,\infty),L^a(\mathbb{R}^n)) u ∈ C (( 0 , ∞ ) , L a ( R n ))
T ( u ) ∈ C ( ( 0 , ∞ ) , L a ( R n ) ) T(u)\in C((0,\infty),L^a(\mathbb{R}^n)) T ( u ) ∈ C (( 0 , ∞ ) , L a ( R n )) for any u ∈ X u\in X u ∈ X t 0 ∈ ( 0 , ∞ ) t_0\in(0,\infty) t 0 ∈ ( 0 , ∞ ) t > t 0 t>t_0 t > t 0
T ( u ) ( t ) = e ( t − t 0 ) Δ T ( u ) ( t 0 ) + ∫ t 0 t e ( t − s ) Δ ( ∣ u ( s ) ∣ b − 1 u ( s ) ) d s = : A ( t ) + B ( t ) . \begin{aligned}
T(u)(t)&=e^{(t-t_0)\Delta}T(u)(t_0)+\int_{t_0}^{t}e^{(t-s)\Delta}(|u(s)|^{b-1}u(s))\,ds\\
&=:A(t)+B(t).
\end{aligned} T ( u ) ( t ) = e ( t − t 0 ) Δ T ( u ) ( t 0 ) + ∫ t 0 t e ( t − s ) Δ ( ∣ u ( s ) ∣ b − 1 u ( s )) d s =: A ( t ) + B ( t ) . Since T ( u ) ( t 0 ) ∈ L a ( R n ) T(u)(t_0)\in L^a(\mathbb{R}^n) T ( u ) ( t 0 ) ∈ L a ( R n ) Lemma 4.10 ) and the heat semigroup is strongly continuous on L a L^a L a A ( t ) ∈ C ( ( t 0 , ∞ ) , L a ( R n ) ) A(t)\in C((t_0,\infty),L^a(\mathbb{R}^n)) A ( t ) ∈ C (( t 0 , ∞ ) , L a ( R n )) B ( t ) B(t) B ( t ) τ > t 0 \tau>t_0 τ > t 0 t > τ t>\tau t > τ
B ( t ) − B ( τ ) = ∫ t 0 t e ( t − τ ) Δ e ( τ − s ) Δ ( ∣ u ( s ) ∣ b − 1 u ( s ) ) d s − B ( τ ) = e ( t − τ ) Δ B ( τ ) − B ( τ ) . \begin{aligned}
B(t)-B(\tau)&=\int_{t_0}^{t}e^{(t-\tau)\Delta}e^{(\tau-s)\Delta}(|u(s)|^{b-1}u(s))\,ds-B(\tau)\\
&=e^{(t-\tau)\Delta}B(\tau)-B(\tau).
\end{aligned} B ( t ) − B ( τ ) = ∫ t 0 t e ( t − τ ) Δ e ( τ − s ) Δ ( ∣ u ( s ) ∣ b − 1 u ( s )) d s − B ( τ ) = e ( t − τ ) Δ B ( τ ) − B ( τ ) . Since sup s ≥ t 0 ∥ u ( s ) ∥ L ∞ ( R n ) ≤ C t 0 − 1 b − 1 \sup_{s\ge t_0}\|u(s)\|_{L^\infty(\mathbb{R}^n)}\le Ct_0^{-\frac{1}{b-1}} sup s ≥ t 0 ∥ u ( s ) ∥ L ∞ ( R n ) ≤ C t 0 − b − 1 1
sup t ≥ t 0 ∥ ∣ u ( s ) ∣ b ∥ L a ≤ C t 0 − 1 sup t ≥ t 0 ∥ u ( s ) ∥ L a ≤ C t 0 − 1 ∥ u 0 ∥ L a . \sup_{t\ge t_0}\||u(s)|^b\|_{L^a}\le Ct_0^{-1}\sup_{t\ge t_0}\|u(s)\|_{L^a}\le Ct_0^{-1}\|u_0\|_{L^a}. t ≥ t 0 sup ∥∣ u ( s ) ∣ b ∥ L a ≤ C t 0 − 1 t ≥ t 0 sup ∥ u ( s ) ∥ L a ≤ C t 0 − 1 ∥ u 0 ∥ L a . and B ( τ ) ∈ L a B(\tau)\in L^a B ( τ ) ∈ L a
∥ B ( t ) − B ( τ ) ∥ L a = ∥ e ( t − τ ) Δ B ( τ ) − B ( τ ) ∥ L a → 0 , \|B(t)-B(\tau)\|_{L^a}=\|e^{(t-\tau)\Delta}B(\tau)-B(\tau)\|_{L^a}\to0, ∥ B ( t ) − B ( τ ) ∥ L a = ∥ e ( t − τ ) Δ B ( τ ) − B ( τ ) ∥ L a → 0 , as t → τ + t \to\tau+ t → τ + L a L^a L a t 0 < t < τ t_0 < t < \tau t 0 < t < τ
∥ B ( t ) − B ( τ ) ∥ L a ≤ ∥ ∫ t 0 t ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s ) ) d s ∥ L a + ∥ ∫ t τ e ( τ − s ) Δ ( ∣ u ( s ) ∣ b − 1 u ( s ) ) d s ∥ L a ≤ ∫ t 0 t ∥ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s ) ) ∥ L a d s + C sup s ≥ t 0 ∥ u ( s ) ∥ L a b ( τ − t ) . (54) \begin{aligned}
\|B(t)-B(\tau)\|_{L^a}
&\leq\left\|\int_{t_0}^{t}\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)(|u(s)|^{b-1}u(s))\,ds\right\|_{L^a}\\
&\quad+\left\|\int_{t}^{\tau}e^{(\tau-s)\Delta}(|u(s)|^{b-1}u(s))\,ds\right\|_{L^a}\\
&\leq\int_{t_0}^{t}\left\|\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)(|u(s)|^{b-1}u(s))\right\|_{L^a}\,ds\\
&\quad+C\sup_{s\geq t_0}\|u(s)\|_{L^a}^{b}(\tau-t). \tag*{(54)}
\end{aligned} ∥ B ( t ) − B ( τ ) ∥ L a ≤ ∫ t 0 t ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s )) d s L a + ∫ t τ e ( τ − s ) Δ ( ∣ u ( s ) ∣ b − 1 u ( s )) d s L a ≤ ∫ t 0 t ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s )) L a d s + C s ≥ t 0 sup ∥ u ( s ) ∥ L a b ( τ − t ) . (54) Given any η > 0 \eta>0 η > 0 γ = γ ( η , t 0 ) \gamma=\gamma(\eta,t_0) γ = γ ( η , t 0 ) 0 < γ < τ − t 0 0<\gamma<\tau-t_0 0 < γ < τ − t 0
γ sup s ≥ t 0 ∥ u ( s ) ∥ L a b ≤ η / 2. \gamma\sup_{s\geq t_0}\|u(s)\|_{L^a}^{b}\leq\eta/2. γ s ≥ t 0 sup ∥ u ( s ) ∥ L a b ≤ η /2. As we will let t → τ − t\to\tau- t → τ − τ − γ < t < τ \tau-\gamma<t<\tau τ − γ < t < τ
∫ t 0 t ∥ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s ) ) ∥ L a d s = ∫ t 0 τ − γ ∥ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s ) ) ∥ L a d s + ∫ τ − γ t ∥ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s ) ) ∥ L a d s ≤ ∫ t 0 τ − γ ∥ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s ) ) ∥ L a d s + 2 γ sup s ≥ t 0 ∥ u ( s ) ∥ L a b . \begin{aligned}
\int_{t_0}^{t}\left\|\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)(|u(s)|^{b-1}u(s))\right\|_{L^a}\,ds
&=\int_{t_0}^{\tau-\gamma}\left\|\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)(|u(s)|^{b-1}u(s))\right\|_{L^a}\,ds\\
&\quad+\int_{\tau-\gamma}^{t}\left\|\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)(|u(s)|^{b-1}u(s))\right\|_{L^a}\,ds\\
&\leq\int_{t_0}^{\tau-\gamma}\left\|\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)(|u(s)|^{b-1}u(s))\right\|_{L^a}\,ds\\
&\quad+2\gamma\sup_{s\geq t_0}\|u(s)\|_{L^a}^{b}.
\end{aligned} ∫ t 0 t ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s )) L a d s = ∫ t 0 τ − γ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s )) L a d s + ∫ τ − γ t ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s )) L a d s ≤ ∫ t 0 τ − γ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ( ∣ u ( s ) ∣ b − 1 u ( s )) L a d s + 2 γ s ≥ t 0 sup ∥ u ( s ) ∥ L a b . Note that for each s ∈ ( t 0 , τ − γ ) s\in(t_0,\tau-\gamma) s ∈ ( t 0 , τ − γ ) ∥ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ∣ u ( s ) ∣ b − 1 u ( s ) ∥ L a → 0 \left\|\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)|u(s)|^{b-1}u(s)\right\|_{L^a}\to0 ( e ( τ − s ) Δ − e ( t − s ) Δ ) ∣ u ( s ) ∣ b − 1 u ( s ) L a → 0 t → τ − t\to\tau- t → τ −
lim sup t → τ − ∫ t 0 t ∥ ( e ( τ − s ) Δ − e ( t − s ) Δ ) ∣ u ( s ) ∣ b − 1 u ( s ) ∥ L a d s ≤ η . \limsup_{t\to\tau-}\int_{t_0}^{t}\left\|\left(e^{(\tau-s)\Delta}-e^{(t-s)\Delta}\right)|u(s)|^{b-1}u(s)\right\|_{L^a}\,ds\leq\eta. t → τ − lim sup ∫ t 0 t ( e ( τ − s ) Δ − e ( t − s ) Δ ) ∣ u ( s ) ∣ b − 1 u ( s ) L a d s ≤ η . Then using this in (54) , we arrive at
lim sup t → τ − ∥ B ( t ) − B ( τ ) ∥ L a ≤ η . \limsup_{t\to\tau-}\|B(t)-B(\tau)\|_{L^a}\leq\eta. t → τ − lim sup ∥ B ( t ) − B ( τ ) ∥ L a ≤ η . Since η > 0 \eta>0 η > 0 B ( t ) ∈ C ( ( t 0 , ∞ ) , L a ) B(t)\in C((t_0,\infty),L^a) B ( t ) ∈ C (( t 0 , ∞ ) , L a )
It remains to show the continuity of u u u L a L^a L a t = 0 t=0 t = 0 a ≥ n ( b − 1 ) 2 a\geq\frac{n(b-1)}{2} a ≥ 2 n ( b − 1 )
lim t → 0 + ∥ T ( u ) ( ⋅ , t ) ∥ L a = 0. (55) \lim_{t\to0+}\|T(u)(\cdot,t)\|_{L^a}=0. \tag*{(55)} t → 0 + lim ∥ T ( u ) ( ⋅ , t ) ∥ L a = 0. (55) Let 1 ≤ p 1 ≤ p 2 ≤ ∞ 1 \le p_1 \le p_2 \le\infty 1 ≤ p 1 ≤ p 2 ≤ ∞
∥ e τ Δ f ∥ L p 2 ≤ C τ − n 2 ( 1 p 1 − 1 p 2 ) ∥ f ∥ L p 1 , τ > 0. \left\|e^{\tau\Delta}f\right\|_{L^{p_2}} \le C\tau^{-\frac{n}{2}\left(\frac{1}{p_1}-\frac{1}{p_2}\right)}\|f\|_{L^{p_1}}, \qquad\tau>0. e τ Δ f L p 2 ≤ C τ − 2 n ( p 1 1 − p 2 1 ) ∥ f ∥ L p 1 , τ > 0. Then if a ≥ b a\ge b a ≥ b
∥ T ( u ) ( ⋅ , t ) ∥ L a ≤ ∫ 0 t ∥ e ( t − s ) Δ ( ∣ u ( s ) ∣ b ) ∥ L a d s ≤ C ∫ 0 t ( t − s ) − n 2 ( b a − 1 a ) ∥ u ( s ) ∥ L a / b b d s ≤ C ( sup 0 < s < t ∥ u ( s ) ∥ L a ) b t 1 − n ( b − 1 ) 2 a . (56) \|T(u)(\cdot,t)\|_{L^a} \le\int_0^t \left\|e^{(t-s)\Delta}(|u(s)|^b)\right\|_{L^a}\,ds
\le C\int_0^t(t-s)^{-\frac{n}{2}\left(\frac{b}{a}-\frac{1}{a}\right)}\|u(s)\|_{L^a/b}^b\,ds
\le C\left(\sup_{0<s<t}\|u(s)\|_{L^a}\right)^b t^{1-\frac{n(b-1)}{2a}}. \tag*{(56)} ∥ T ( u ) ( ⋅ , t ) ∥ L a ≤ ∫ 0 t e ( t − s ) Δ ( ∣ u ( s ) ∣ b ) L a d s ≤ C ∫ 0 t ( t − s ) − 2 n ( a b − a 1 ) ∥ u ( s ) ∥ L a / b b d s ≤ C ( 0 < s < t sup ∥ u ( s ) ∥ L a ) b t 1 − 2 a n ( b − 1 ) . (56) On the other hand, if a < b a<b a < b
∥ T ( u ) ( ⋅ , t ) ∥ L a ≤ ∫ 0 t ∥ e ( t − s ) Δ ( ∣ u ( s ) ∣ b ) ∥ L a d s ≤ C ∫ 0 t ( t − s ) − n 2 ( 1 − 1 a ) ∥ u ( s ) b ∥ L 1 d s ≤ C ( sup τ > 0 τ 1 b − 1 ∥ u ( τ ) ∥ L ∞ ) b − a ( sup 0 < s < t ∥ u ( s ) ∥ L a ) a × ∫ 0 t ( t − s ) − n 2 ( 1 − 1 a ) s − b − a b − 1 d s . \begin{aligned}
\|T(u)(\cdot,t)\|_{L^a} \le\int_0^t \left\|e^{(t-s)\Delta}(|u(s)|^b)\right\|_{L^a}\,ds
\le C\int_0^t(t-s)^{-\frac{n}{2}\left(1-\frac{1}{a}\right)}\|u(s)^b\|_{L^1}\,ds \\
&\le C\left(\sup_{\tau>0}\tau^{\frac{1}{b-1}}\|u(\tau)\|_{L^\infty}\right)^{b-a}
\left(\sup_{0<s<t}\|u(s)\|_{L^a}\right)^a
\times\int_0^t(t-s)^{-\frac{n}{2}\left(1-\frac{1}{a}\right)}s^{-\frac{b-a}{b-1}}\,ds.
\end{aligned} ∥ T ( u ) ( ⋅ , t ) ∥ L a ≤ ∫ 0 t e ( t − s ) Δ ( ∣ u ( s ) ∣ b ) L a d s ≤ C ∫ 0 t ( t − s ) − 2 n ( 1 − a 1 ) ∥ u ( s ) b ∥ L 1 d s ≤ C ( τ > 0 sup τ b − 1 1 ∥ u ( τ ) ∥ L ∞ ) b − a ( 0 < s < t sup ∥ u ( s ) ∥ L a ) a × ∫ 0 t ( t − s ) − 2 n ( 1 − a 1 ) s − b − 1 b − a d s . This gives
∥ T ( u ) ( ⋅ , t ) ∥ L a ≤ C ( ∥ τ 1 b − 1 u ( τ ) ∥ L τ ∞ L x ∞ ) b − a ( sup 0 < s < t ∥ u ( s ) ∥ L a ) a t ( 1 − 1 a ) ( − n 2 + a b − 1 ) . (57) \|T(u)(\cdot,t)\|_{L^a} \le C\left(\|\tau^{\frac{1}{b-1}}u(\tau)\|_{L^\infty_\tau L^\infty_x}\right)^{b-a}
\left(\sup_{0<s<t}\|u(s)\|_{L^a}\right)^a
t^{\left(1-\frac{1}{a}\right)\left(-\frac{n}{2}+\frac{a}{b-1}\right)}. \tag*{(57)} ∥ T ( u ) ( ⋅ , t ) ∥ L a ≤ C ( ∥ τ b − 1 1 u ( τ ) ∥ L τ ∞ L x ∞ ) b − a ( 0 < s < t sup ∥ u ( s ) ∥ L a ) a t ( 1 − a 1 ) ( − 2 n + b − 1 a ) . (57) Thus, it follows from (56) and (57) that equality (55) holds in the case a > n ( b − 1 ) 2 a>\frac{n(b-1)}{2} a > 2 n ( b − 1 )
Finally, we consider the case a = n ( b − 1 ) 2 a=\frac{n(b-1)}{2} a = 2 n ( b − 1 ) u N → u u_N\to u u N → u L t ∞ L x a L^\infty_tL^a_x L t ∞ L x a u N , u ∈ C ( ( 0 , ∞ ) , L a ) u_N,u\in C((0,\infty),L^a) u N , u ∈ C (( 0 , ∞ ) , L a ) u N ( 0 ) = u ( 0 ) = u 0 u_N(0)=u(0)=u_0 u N ( 0 ) = u ( 0 ) = u 0 N ∈ N N\in\mathbb{N} N ∈ N u N u_N u N t = 0 t=0 t = 0 N ∈ N N\in\mathbb{N} N ∈ N u 1 = e t Δ u 0 u_1=e^{t\Delta}u_0 u 1 = e t Δ u 0
lim t → 0 + ∥ T ( v ) ( ⋅ , t ) ∥ L a = 0 (58) \lim_{t\to0+}\|T(v)(\cdot,t)\|_{L^a}=0 \tag*{(58)} t → 0 + lim ∥ T ( v ) ( ⋅ , t ) ∥ L a = 0 (58) for any v ∈ L ∞ ( ( 0 , ∞ ) , L a ) ∩ C ( [ 0 , ∞ ) , L a ) v\in L^\infty((0,\infty),L^a)\cap C([0,\infty),L^a) v ∈ L ∞ (( 0 , ∞ ) , L a ) ∩ C ([ 0 , ∞ ) , L a ) v ( 0 ) = u 0 v(0)=u_0 v ( 0 ) = u 0 τ 1 b − 1 v ( τ ) ∈ L ∞ ( ( 0 , ∞ ) , L ∞ ( R n ) ) \tau^{\frac{1}{b-1}}v(\tau)\in L^\infty((0,\infty),L^\infty(\mathbb{R}^n)) τ b − 1 1 v ( τ ) ∈ L ∞ (( 0 , ∞ ) , L ∞ ( R n )) η > 0 \eta>0 η > 0 ϕ ∈ C c ∞ ( R n ) \phi\in C^\infty_c(\mathbb{R}^n) ϕ ∈ C c ∞ ( R n )
∥ u 0 − ϕ ∥ L a ( R n ) ≤ η \|u_0-\phi\|_{L^a(\mathbb{R}^n)}\le\eta ∥ u 0 − ϕ ∥ L a ( R n ) ≤ η and then by the continuity of v v v t = 0 t=0 t = 0 s ∗ > 0 s^*>0 s ∗ > 0
sup 0 < s < s ∗ ∥ v ( s ) − ϕ ∥ L a ( R n ) ≤ 2 η . \sup_{0<s<s^*}\|v(s)-\phi\|_{L^a(\mathbb{R}^n)}\le2\eta. 0 < s < s ∗ sup ∥ v ( s ) − ϕ ∥ L a ( R n ) ≤ 2 η . We now write
∥ T ( v ) ( ⋅ , t ) ∥ L a ≤ C ∫ 0 t ∥ e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) ∥ L a d s + C ∫ 0 t ∥ e ( t − s ) Δ ( ∣ ϕ ∣ b ) ∥ L a d s ≤ C ∫ 0 t ∥ e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) ∥ L a d s + C t ∥ ϕ b ∥ L a . (59) \begin{aligned}
\|T(v)(\cdot,t)\|_{L^a} &\le C\int_0^t \left\|e^{(t-s)\Delta}(|v(s)-\phi|^b)\right\|_{L^a}\,ds \\
&\quad+ C\int_0^t \left\|e^{(t-s)\Delta}(|\phi|^b)\right\|_{L^a}\,ds \\
&\le C\int_0^t \left\|e^{(t-s)\Delta}(|v(s)-\phi|^b)\right\|_{L^a}\,ds + Ct\|\phi^b\|_{L^a}. \tag*{(59)}
\end{aligned} ∥ T ( v ) ( ⋅ , t ) ∥ L a ≤ C ∫ 0 t e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) L a d s + C ∫ 0 t e ( t − s ) Δ ( ∣ ϕ ∣ b ) L a d s ≤ C ∫ 0 t e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) L a d s + C t ∥ ϕ b ∥ L a . (59) Note that as in (56) and (57) with a = n ( b − 1 ) 2 a=\frac{n(b-1)}{2} a = 2 n ( b − 1 ) 0 < t < s ∗ 0<t<s^* 0 < t < s ∗
∫ 0 t ∥ e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) ∥ L a d s ≤ C ( sup 0 < s < t ∥ v ( s ) − ϕ ∥ L a ) b ≤ C ( 2 η ) b \int_0^t \left\|e^{(t-s)\Delta}(|v(s)-\phi|^b)\right\|_{L^a}\,ds
\le C\left(\sup_{0<s<t}\|v(s)-\phi\|_{L^a}\right)^b
\le C(2\eta)^b ∫ 0 t e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) L a d s ≤ C ( 0 < s < t sup ∥ v ( s ) − ϕ ∥ L a ) b ≤ C ( 2 η ) b if a ≥ b a\ge b a ≥ b
∫ 0 t ∥ e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) ∥ L a d s ≤ C ( ∥ τ 1 b − 1 v ( τ ) ∥ L τ ∞ L x ∞ ) b − a ( sup 0 < s < t ∥ v ( s ) − ϕ ∥ L a ) a ≤ C ( ∥ τ 1 b − 1 v ( τ ) ∥ L τ ∞ L x ∞ ) b − a ( 2 η ) a \begin{aligned}
\int_0^t \left\|e^{(t-s)\Delta}(|v(s)-\phi|^b)\right\|_{L^a}\,ds
&\le C\left(\|\tau^{\frac{1}{b-1}}v(\tau)\|_{L^\infty_\tau L^\infty_x}\right)^{b-a}
\left(\sup_{0<s<t}\|v(s)-\phi\|_{L^a}\right)^a \\
&\le C\left(\|\tau^{\frac{1}{b-1}}v(\tau)\|_{L^\infty_\tau L^\infty_x}\right)^{b-a}(2\eta)^a
\end{aligned} ∫ 0 t e ( t − s ) Δ ( ∣ v ( s ) − ϕ ∣ b ) L a d s ≤ C ( ∥ τ b − 1 1 v ( τ ) ∥ L τ ∞ L x ∞ ) b − a ( 0 < s < t sup ∥ v ( s ) − ϕ ∥ L a ) a ≤ C ( ∥ τ b − 1 1 v ( τ ) ∥ L τ ∞ L x ∞ ) b − a ( 2 η ) a if a < b a<b a < b
As η \eta η (59) , we see that limit (58) holds as desired. □
Acknowledgements. G.Y.D. is supported in part by a start-up grant from the University of Toronto, and NSERC grants RGPIN-2018-06487 and RGPIN-2025-06419. N. C. Phuc is supported in part by Simons Foundation [Record ID: MPS-TSM-00002686].
AI Usage Disclosure. ChatGPT Plus GPT-5.6 Sol was used to show the continuity in time of T ( u ) T(u) T ( u ) Theorem 4.11 . The authors take full responsibility for the content of the paper. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada.
Email address: gaely.diebou@utoronto.ca
Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803, USA.
Email address: pcnguyen@math.lsu.edu
