Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces

Abstract

We present the first to date weighted and mixed-norm Sobolev estimates for fully nonlinear elliptic and parabolic equations in the whole space under a relaxed convexity condition with almost VMO dependence on space-time variables. The corresponding interior and boundary estimates are also obtained.

Appendix

Here we take $\Omega =\Omega ^{1}\times \cdots \times \Omega ^{d}$, where $\Omega ^j={\mathbb {R}}$ or ${\mathbb {R}}_+$, $j=1,\ldots ,d$ and let $\mu$ to be the Lebesgue measure on $\Omega$. We take integers $0= l_{0}<l_{1}<\ldots <l_{m}=d$ and express points in $\Omega$ as

$$\begin{aligned} x=(x_{1},\ldots ,x_{d})=({\check{x}}_{1},\ldots ,{\check{x}}_{m}), \end{aligned}$$

where ${\check{x}}_{i}=(x_{l_{i-1}+1},\ldots , x_{l_{i}})$ and set

$$\begin{aligned} {\check{\Omega }}^{i}=\Omega ^{l_{i-1}+1} \times \cdots \times \Omega ^{l_{i}},\quad {\hat{\Omega }}^{i}= \Omega ^{l_{i-1}+1}\times \cdots \times \Omega ^{d}, \end{aligned}$$

${\hat{x}}_{i}=(x_{l_{i}+1},\ldots ,x_{d})$. Take $k(1),\ldots ,k(d)\in \{1,2,\ldots \}$ and, for $n\in {\mathbb {Z}}$, let

$$\begin{aligned} {\check{C}}_{n}^{i}=[0,2^{-nk(l_{i-1}+1)})\times \cdots \times [0,2^{-nk(l_{i})}) \end{aligned}$$

be a subset of ${\check{\Omega }}^{i}$ and $C_{n}={\check{C}}_{n}^{1}\times \cdots \times {\check{C}}_{n}^{m}$. By $A_{p}$-weights on ${\check{\Omega }}^{i}$ we mean the $A_{p}$-weights relative to all translates of ${\check{C}}_{n}^{i}$, $n\in {\mathbb {Z}}$, belonging to ${\check{\Omega }}^{i}$, and, naturally, $A_{p}$-weights on $\Omega$ are defined using all translates of $C_{n}$, $n\in {\mathbb {Z}}$, belonging to $\Omega$.

Theorem 7.11

Let $K_0,p_{k}\in (1,\infty )$, $w^k\in A_{p_{k}}( {\check{\Omega }}^k)$, $[w^k]_{p_{k}}\le K_0$, $k=1,\ldots ,m$, and u, g be measurable functions on $\Omega$. Then there exists a constant $\Lambda _0=\Lambda _0(d,p_1,\ldots ,p_{m}, k(1), \ldots ,k(d),K_0)\ge 1$ such that if

$$\begin{aligned} \Vert u\Vert _{L_{p_{1}}(w\,d\mu )} \le N_0\Vert g\Vert _{L_{p_{1}}(w\,d\mu )} \end{aligned}$$

for some $N_0\in (0,\infty )$ and for every $w\in A_{p_{1}}(\Omega )$ with $[w]_{p_{1}}\le \Lambda _0$, then we have

$$\begin{aligned} \Vert u\Vert _{L_{p_{1},\ldots ,p_{m}}(w^1,\ldots ,w^{m})}\le N\Vert g\Vert _{L_{p_{1},\ldots ,p_{m}}(w^1,\ldots ,w^m)}, \end{aligned}$$

where the norms are defined as in (3.18) replacing $dx_{i}$ by $w^{i}({\check{x}}_{i})\,d{\check{x}}_{i}$, the constant N depends only on d, $p_1,\ldots ,p_m, k(1),\ldots ,k(d)$, $K_0$, and $N_0$.

Proof

We follow the proof of Corollary 2.7 in [11]. Recall the extrapolation theorem of Rubio de Francia [24] which says that for any constant $\Lambda _{j }\in (1, \infty )$, $j= 1,\ldots ,m$, there exists a constant $\Lambda _{j-1}=\Lambda _{j-1}(d-j,p_{j},p_{j+1}, K_{0} \Lambda _{j } )\in ( 1, \infty )$ (we drop its dependence on the k(i)’s) such that, if

• (a) for two nonnegative functions $U_{j}$ and $G_{j}$ on ${\hat{\Omega }}^{j+1}$,

  $$\begin{aligned} \int _{{\hat{\Omega }}^{j+1} } U_{j}^{p_{j}}w({\hat{x}}_{j+1} )\, d{\hat{x}}_{j+1} \le N_{j}\int _{{\hat{\Omega }}^{j+1} } G_{j}^{p_{j}}w({\hat{x}}_{j+1} )\, d{\hat{x}}_{j+1} \end{aligned}$$

  for some $N_j\in (0,\infty )$ and for every $w\in A_{p_{j}}({\hat{\Omega }}^{j+1} )$ with $[w]_{p_{j}}\le \Lambda _{j-1}$, then

• (b) we have

  $$\begin{aligned} \int _{{\hat{\Omega }}^{j+1} } U_{j}^{p_{j+1}}w({\hat{x}}_{j+1} )\, d{\hat{x}}_{j+1} \le N_{j+1}\int _{{\hat{\Omega }}^{j+1} } G_{j}^{p_{j+1}}w({\hat{x}}_{j+1} )\, d{\hat{x}}_{j+1} \end{aligned}$$

  (7.3)

  for some $N_{j+1}\in (0,\infty )$, depending only on d, j, $K_0\Lambda _j$, $p_{j}$, $p_{j+1}$, and $N_{j}$, and for every $w\in A_{p_{j+1}}({\hat{\Omega }}^{j+1} )$ with $[w]_{p_{j+1}}\le K_{0}\Lambda _{j}$.

$\square$

In this form the theorem is proved in [11]. We define $\Lambda _{m-1}=1$ and find all $\Lambda _{j}$, $j=0,1,\ldots ,m-1$. Then assume that $m\ge 2$ and define $U_{0}(x)=u(x)$,

$$\begin{aligned} U_{j}({\hat{x}}_{j+1})= \left( \int _{{\check{\Omega }}^{j}}U^{p_{j}}_{j-1}({\hat{x}}_{j}) \, w^{j}({\check{x}}_{j})\,d{\check{x}}_{j}\right) ^{1/p_{j}},\quad 1\le j\le m-1, \end{aligned}$$

and similarly we introduce $G_{j}$’s by taking g in place of u. To prove the theorem, it suffices to prove that (b) holds for $j=m-1$ because $w^{m}\in A_{p_{m}}({\check{\Omega }}^{m})$ and $[w^{ m}]_{p_{m}}\le K_{0} = K_{0}\Lambda _{m-1}$. We are going to use the induction on $j=0,1,\ldots ,m-1$.

Observe that (b) holds for $j=0$ by assumption. Suppose that it holds for a $j\in \{0,1,\ldots ,m-2\}$. Then (7.3) also holds for

$$\begin{aligned} w({\hat{x}}_{j+1} ):=w^{j+1}({\check{x}}_{j+1})w({\hat{x}}_{j+2} ) \end{aligned}$$

if $w^{j+1}\in A_{p_{j+1}}( {\check{\Omega }}^{j+1})$ and $w({\hat{x}}_{j+2} ) \in A_{p_{j+1}}({\hat{\Omega }}^{j+2})$ with

$$\begin{aligned}{}[w^{j+1}]_{p_{j+1}}\le K_{0},\quad [w({\hat{x}}_{j+2} )]_{p_{j+1}}\le \Lambda _{j } \end{aligned}$$

because then $[w({\hat{x}}_{j+1} )]_{p_{j+1}}\le K_{0}\Lambda _{j }$. Remarkably, this implies that (a) holds with $j+1$ in place of j. Then (b) also holds with $j+1$ in place of j. This justifies the induction and proves the theorem.