Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

Abstract

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class $A_{q}$ -weights $\begin{matrix} u_{t} - div (A (x, t) \nabla u) = div (F), \end{matrix}$ in a bounded domain $Ω \times (0, T) \subset R^{N + 1}$ , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

Keywords

Quasilinear parabolic equations maximal potential Reifenberg flat domain

1. Introduction and main results

In this article, we are concerned with the global weighted Lorentz space estimates for gradients of very weak solutions to linear parabolic equations in the divergence form: $\begin{matrix} (1.1) & \{\begin{matrix} u_{t} - div (A (x, t) \nabla u) = div (F) & in Ω_{T}, \\ u = 0 & on \partial_{p} (Ω \times (0, T)), \end{matrix} \end{matrix}$ where $Ω_{T} : = Ω \times (0, T)$ is a bounded open subset of $R^{N + 1}$ , $N ⩾ 2$ , $\partial_{p} (Ω \times (0, T)) = (\partial Ω \times (0, T)) \cup (Ω \times {t = 0})$ , $F \in L^{p} (Ω_{T}, R^{N})$ , $p > 1$ is a given vector field and the matrix function $A : R^{N} \times R \to R^{N^{2}}$ is a measurable function in $(x, t)$ . We also assume that A is uniformly elliptic, i.e, $\begin{matrix} (1.2) & Λ^{- 1} | ξ |^{2} ⩽ ⟨ A (x, t) ξ, ξ ⟩ ⩽ Λ | ξ |^{2}, \end{matrix}$ for every $ξ \in R^{N}$ and a.e. $(x, t) \in R^{N} \times R$ , where Λ is a positive constant. Our main result is that, for any $q > 1$ and any $w \in A_{q}$ (the parabolic Muckenhoupt class, see below), $F \in L_{w}^{q} (Ω_{T}, R^{N})$ , and under some additional conditions on the matrix A and on the boundary of Ω, there exists a unique very weak solution $u \in L^{q_{0}} (0, T, W_{0}^{1, q_{0}} (Ω))$ for some $q_{0} > 1$ of the problem (1.1) satisfying $\begin{array}{l} (1.3) & \int_{Ω_{T}} | \nabla u |^{q} w d x d t ⩽ C \int_{Ω_{T}} | F |^{q} w d x d t . \end{array}$ In this paper, a very weak solution u of (1.1) is understood in the standard weak (distributional) sense, that is $u \in L^{1} (0, T, W_{0}^{1, 1} (Ω))$ is called a very weak solution of (1.1) if $\begin{array}{l} - \int_{Ω_{T}} u φ_{t} d x d t + \int_{Ω_{T}} A (x, t) \nabla u \nabla φ d x d t = - \int_{Ω_{T}} F \nabla φ d x d t \end{array}$ for all $φ \in C_{c}^{1} ([0, T) \times Ω)$ .

In the case $w \equiv 1$ , the analogous result was obtained by Byun and Wang in [2,3]. For the nonconstant weights $w \in A_{q / 2}$ with $q > 2$ and for any $ϵ \in (0, 1)$ , $w^{2 + ϵ} \in A_{1}$ with $\frac{2 (1 + ϵ)}{2 + ϵ} < p ⩽ 2$ , the estimate (1.3) was proved by the second named author in [10, see Theorem 1.2, 1.3]. The emphasis of the paper is obtaining the tighter estimate (1.3) for all $w \in A_{q}$ , $q > 1$ , which is much more difficult to obtain. The study of this work is motivated by [1] where they have demonstrated for linear elliptic equation with $w \in A_{q}$ , $q > 1$ , their approach employs a local version of the sharp maximal function of Fefferman and Stein (see [16]). Our approach in this paper is different from that of [1,5], we mainly take advantage of the Hardy-Littlewood maximal function and use (2.13). It is worth mentioning that our results can imply the results in [1,5], see Remark 1.2 below. Furthermore, the requirement $w \in A_{q}$ in (1.3) is optimal, that was discussed in [1,16,17].

Throughout the paper, we need to assume that Ω is a Lipschitz domain with small Lipschitz constant. We say that Ω is a $(δ, R_{0})$ -Lip domain for $δ \in (0, 1)$ and $R_{0} > 0$ if for every $x \in \partial Ω$ , there exists a map $Γ : R^{n - 1} \to R$ such that $‖ \nabla Γ ‖_{L^{\infty} (R^{n - 1})} ⩽ δ$ and, upon rotating and relabeling of coordinates if necessary, $\begin{array}{l} Ω \cap B_{R_{0}} (x_{0}) = {(x^{'}, x_{N}) \in B_{R_{0}} (x_{0}) : x_{N} > Γ (x^{'})} . \end{array}$ It is well-known that Ω is a $(δ, R_{0})$ -Lip domain for $δ \in (0, 1)$ and $R_{0} > 0$ , then Ω is also a $(δ, R_{0})$ -Reifenberg flat domain, see [2,3,10]. We also require that the matrix function A satisfies a smallness condition of BMO type in the x-variable in the sense that $A (x, t)$ satisfies a $(δ, R_{0})$ -BMO condition for some $δ, R_{0} > 0$ if $\begin{matrix} {[A]}_{R_{0}} : = sup_{(y, s) \in R^{N + 1}, r ⩽ R_{0}} ⨏_{Q_{r} (y, s)} | A (x, t) - {\overline{A}}_{B_{r} (y)} (t) | d x d t ⩽ δ, \end{matrix}$ where $Q_{r} (y, s) = B_{r} (y) \times (s - r^{2}, s)$ and ${\overline{A}}_{B_{r} (y)} (t)$ is denoted the average of $A (\cdot, t)$ over the ball $B_{r} (y)$ , i.e, ${\overline{A}}_{B_{r} (y)} (t) : = ⨏_{B_{r} (y)} A (x, t) d x$ .

The above condition has been appeared in [1,10,11]. It is easy to see that the $(δ, R_{0})$ -BMO is satisfied when A is continuous or has small jump discontinuities with respect to x. We recall that a positive function $w \in L_{loc}^{1} (R^{N + 1})$ is called an $A_{p}$ weight, $1 ⩽ p < \infty$ if there holds $\begin{array}{l} {[w]}_{A_{p}} : = sup_{{\tilde{Q}}_{ρ} (x, t)} (⨏_{{\tilde{Q}}_{ρ} (x, t)} w (y, s) d y d s) {(⨏_{{\tilde{Q}}_{ρ} (x, t)} w {(y, s)}^{- \frac{1}{p - 1}} d y d s)}^{p - 1} < \infty if p > 1, \\ {[w]}_{A_{1}} : = sup_{{\tilde{Q}}_{ρ} (x, t)} (⨏_{{\tilde{Q}}_{ρ} (x, t)} w (y, s) d y d s) {‖ w^{- 1} ‖}_{L^{\infty} ({\tilde{Q}}_{ρ} (x, t))} < \infty, \end{array}$ with ${\tilde{Q}}_{ρ} (x, t) = B_{ρ} (x) \times (t - ρ^{2} / 2, t + ρ^{2} / 2) \subset R^{N + 1}$ . The quantity ${[w]}_{A_{p}}$ is called the $A_{p}$ constant of w.

A positive function $w \in L_{loc}^{1} (R^{N + 1})$ is called an $A_{\infty}$ weight if there are two positive constants C and ν such that $\begin{matrix} w (E) ⩽ C {(\frac{| E |}{| Q |})}^{ν} w (Q), \end{matrix}$ for all cylinders $Q = Q_{ρ} (x, t)$ and all measurable subsets E of Q. The pair $(C, ν)$ is called the $A_{\infty}$ constant of w and is denoted by ${[w]}_{A_{\infty}}$ . It is well known that this class is the union of $A_{p}$ for all $p \in (1, \infty)$ , see [7]. Furthermore, if $w \in A_{p}$ with ${[w]}_{A_{p}} ⩽ M$ , then there exist a constant $ε_{0} = ε (N, p, M)$ , and a constant $M_{0} = M (N, p, M)$ such that ${[w]}_{A_{p - ε_{0}}} ⩽ M_{0}$ . If w is a weight function belonging to $A_{\infty}$ and $E \subset R^{N + 1}$ a Borel set, $0 < q < \infty$ , $0 < p ⩽ \infty$ , the weighted Lorentz space $L_{w}^{q, p} (E)$ is the set of measurable functions g on E such that $\begin{matrix} ‖ g ‖_{L_{w}^{q, p} (E)} : = \{\begin{matrix} {(q \int_{0}^{\infty} {(ρ^{q} w ({(x, t) \in E : | g (x, t) | > ρ}))}^{\frac{p}{q}} \frac{d ρ}{ρ})}^{1 / p} < \infty & if s < \infty, \\ {sup}_{ρ > 0} ρ {(w ({(x, t) \in E : | g (x, t) | > ρ}))}^{1 / q} < \infty & if p = \infty . \end{matrix} \end{matrix}$ Here we write $w (O) = \int_{O} w (x, t) d x d t$ for a measurable set $O \subset R^{N + 1}$ . We always denote $T_{0} = diam (Ω) + T^{1 / 2}$ and $Q_{ρ} (x, t) = B_{ρ} (x) \times (t - ρ^{2}, t)$ , ${\tilde{Q}}_{ρ} (x, t) = B_{ρ} (x) \times (t - ρ^{2} / 2, t + ρ^{2} / 2)$ for $(x, t) \in R^{N + 1}$ and $ρ > 0$ . Moreover, $M$ denotes the parabolic Hardy-Littlewood maximal function defined for each locally integrable function f on $R^{N + 1}$ by $\begin{matrix} M (f) (x, t) = sup_{ρ > 0} ⨏_{{\tilde{Q}}_{ρ} (x, t)} | f (y, s) | d y d s \forall (x, t) \in R^{N + 1} . \end{matrix}$ If $q > 1$ and $w \in A_{q}$ , we verify that $M$ is a bounded operator from $L^{1} (R^{N + 1})$ into $L^{1, \infty} (R^{N + 1})$ and from $L_{w}^{q, p} (R^{N + 1})$ into itself for $0 < p ⩽ \infty$ , see [16,17].

The main result of this paper is based on a new global good-λ type inequality that involves the Hardy-Littlewood maximal function (see Theorem 3.1). We would like to mention that a variant of this kind of global good-λ type inequality can also be found in certain nonlinear measure datum problems (see [12–15]). We now state the main result of the paper.

Theorem 1.1.
For any $w \in A_{q}$ , $1 < q < \infty$ , $0 < p ⩽ \infty$ we find $δ = δ (N, Λ, q, p, {[w]}_{A_{q}}) \in (0, 1)$ such that if Ω is $(δ, R_{0})$ -Lip domain Ω and ${[A]}_{R_{0}} ⩽ δ$ for some $R_{0} > 0$ and $F \in L_{w}^{q, p} (Ω_{T})$ , then there exists a unique weak solution $u \in L^{q_{0}} (0, T, W_{0}^{1, q_{0}} (Ω))$ for some $q_{0} > 1$ of the problem ( 1.1 ) satisfying $\begin{matrix} (1.4) & | ‖ \nabla u | ‖_{L_{w}^{q, p} (Ω_{T})} ⩽ C ‖ F ‖_{L_{w}^{q, p} (Ω_{T})} . \end{matrix}$ Here C depends only on N, Λ, q, p, ${[w]}_{A_{q}}$ and $T_{0} / R_{0}$ .

We can get a version of Theorem 1.1 for the linear elliptic equations which was obtained in [1]. More precisely, using the same method in this paper we can prove the following result. Remark 1.2.
Assume that $A (x) = A (x, t)$ for all $(x, t) \in R^{N + 1}$ . For any $w (x) = w (x, t) \in A_{q}$ , $1 < q < \infty$ , $0 < p ⩽ \infty$ , we find $δ = δ (N, Λ, q_{0}, q, p, {[w]}_{A_{q}}) \in (0, 1)$ such that if Ω is $(δ, R_{0})$ -Lip domain Ω and ${[A]}_{R_{0}} ⩽ δ$ for some $R_{0} > 0$ and $G \in L_{w}^{q, p} (Ω)$ , then there exists a unique very weak solution $u \in W_{0}^{1, q_{0}} (Ω)$ for some $q_{0} > 1$ of $\begin{matrix} (1.5) & \{\begin{matrix} - div (A (x) \nabla u) = div (G) & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ satisfying $\begin{matrix} (1.6) & | ‖ \nabla u | ‖_{L_{w}^{q, p} (Ω)} ⩽ C ‖ G ‖_{L_{w}^{q, p} (Ω)} . \end{matrix}$ Here C depends only on N, Λ, q, p, ${[w]}_{A_{q}}$ and $diam (Ω) / R_{0}$ .

2. Interior estimates and boundary estimates for parabolic equations

In this section, we present various local interior and boundary estimates for the very weak solution u of (1.1). They will be used for our global estimates later. In [10], the author proved the following result.

Theorem 2.1.
Let $q > 1$ and $G \in L^{q} (Ω_{T}, R^{N})$ . We find a $δ = δ (N, Λ, q) \in (0, 1)$ such that if Ω is a $(δ, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ$ for some $R_{0} > 0$ , then there exists a unique very weak solution $u \in L^{q} (0, T, W_{0}^{1, q} (Ω))$ of $\begin{matrix} (2.1) & \{\begin{matrix} u_{t} - div (A (x, t) \nabla u) = div (G) & in Ω_{T}, \\ u = 0 & on \partial_{p} (Ω \times (0, T)) . \end{matrix} \end{matrix}$ Furthermore, there holds $\begin{matrix} (2.2) & ‖ \nabla u ‖_{L^{q} (Ω_{T})} ⩽ C ‖ F ‖_{L^{q} (Ω_{T})} \end{matrix}$ where C depends only on N, Λ, q and $(diam (Ω) + T^{1 / 2}) / R_{0}$ .

Let $s > 1$ , we apply Theorem 2.1 to $G = F$ and $q = s$ , there is a constant $δ_{0} = δ_{0} (N, Λ, s) \in (0, 1 / 4)$ such that if Ω is a $(δ_{0}, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ_{0}$ for some $R_{0} > 0$ , then the problem (1.1) has a unique very weak solution $u \in L^{s} (0, T, W_{0}^{1, s} (Ω))$ satisfying $\begin{matrix} ‖ \nabla u ‖_{L^{s} (Ω_{T})} ⩽ C ‖ F ‖_{L^{s} (Ω_{T})} . \end{matrix}$ where $C = C (N, Λ, s, T_{0} / R_{0})$ .

In this section, we assume that Ω is a $(δ_{0}, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ_{0}$ for some $R_{0} > 0$ , where $δ_{0}$ is as above. For some technical reasons, we always assume that $u \in L^{2} (- \infty, T; W_{0}^{1, 2} (Ω))$ is a very weak solution to the problem (1.1) in $Ω \times (- \infty, T)$ with $F = 0$ in $Ω \times (- \infty, 0)$ and $F \in L^{2} (Ω_{T}) .$
2.1. Interior estimates

Let $R \in (0, R_{0})$ , $B_{2 R} = B_{2 R} (x_{0}) \subset \subset Ω$ and $t_{0} \in (0, T)$ . Set $Q_{2 R} = B_{2 R} \times (t_{0} - 4 R^{2}, t_{0})$ and $\partial_{p} Q_{2 R} = (\partial B_{2 R} \times (t_{0} - 4 R^{2}, t_{0})) \cup (B_{2 R} \times {t = t_{0} - 4 R^{2}})$ . Since ${[A]}_{R} ⩽ δ_{0}$ , thus, applying Theorem 2.1 to $Ω_{T} = Q_{2 R}$ and $G = F$ , the following problem $\begin{matrix} (2.3) & \{\begin{matrix} W_{t} - div (A (x, t) \nabla W) = div (F) & in Q_{2 R}, \\ W = 0 & on \partial_{p} Q_{2 R}, \end{matrix} \end{matrix}$ has a unique very weak solution $W \in L^{2} (t_{0} - 4 R^{2}, t_{0}; W_{0}^{1, 2} (B_{2 R}))$ . Moreover, we have $\begin{array}{l} (2.4) & ⨏_{Q_{2 R}} | \nabla W |^{s} d x d t ⩽ C ⨏_{Q_{2 R}} | F |^{s} d x d t \forall s > 1, \end{array}$ where C depends only on N, Λ, s. Note that the constant $(diam (Ω) + T^{1 / 2}) / R_{0}$ in Theorem 2.1 equals 6 in this case.

We now set $w = u - W$ , so $w \in L^{2} (t_{0} - 4 R^{2}, t_{0}; W^{1, 2} (B_{2 R}))$ is a weak solution of $\begin{matrix} (2.5) & w_{t} - div (A (x, t) \nabla w) = 0 in Q_{2 R} . \end{matrix}$ We need the following a variant of Gehring’s lemma which was proved in [6,9].

Lemma 2.2.
There exists a constant $C > 0$ depending only on N, Λ such that the following estimate $\begin{matrix} (2.6) & {(⨏_{Q_{ρ / 2} (y, s)} | \nabla w |^{2} d x d t)}^{\frac{1}{2}} ⩽ C ⨏_{Q_{ρ} (y, s)} | \nabla w | d x d t, \end{matrix}$ holds for all $Q_{ρ} (y, s) \subset \subset Q_{2 R}$ .

In the next, we denote by v the unique solution $v \in L^{2} (t_{0} - R^{2}, t_{0}; W^{1, 2} (B_{R}))$ of the following problem $\begin{matrix} (2.7) & \{\begin{matrix} v_{t} - div ({\overline{A}}_{B_{R}} (t) \nabla v) = 0 & in Q_{R}, \\ v = w & on \partial_{p} Q_{R}, \end{matrix} \end{matrix}$ where $B_{R} = B_{R} (x_{0})$ , $Q_{R} = B_{R} \times (t_{0} - R^{2}, t_{0})$ and $\begin{matrix} \partial_{p} Q_{R} = (\partial B_{R} \times (t_{0} - R^{2}, t_{0})) \cup (B_{R} \times {t = t_{0} - R^{2}}) . \end{matrix}$ Lemma 2.3.
There exists a constant $C = C (N, Λ) > 0$ such that $\begin{matrix} (2.8) & {(⨏_{Q_{R}} {| \nabla (w - v) |}^{2} d x d t)}^{1 / 2} ⩽ C {[A]}_{R_{0}} ⨏_{Q_{2 R}} | \nabla w | d x d t, \end{matrix}$ and $\begin{matrix} (2.9) & C^{- 1} \int_{Q_{R}} | \nabla v |^{2} d x d t ⩽ \int_{Q_{R}} | \nabla w |^{2} d x d t ⩽ C \int_{Q_{R}} | \nabla v |^{2} d x d t . \end{matrix}$
Proof.
The proof can be found in [11, Lemma 7.3]. □
Proposition 2.4.
There holds $\begin{matrix} v \in L^{2} (t_{0} - R^{2}, t_{0}; W^{1, 2} (B_{R})) \cap L^{\infty} (t_{0} - \frac{1}{4} R^{2}, t_{0}; W^{1, \infty} (B_{R / 2})), \end{matrix}$ and $\begin{array}{l} (2.10) & ‖ \nabla v ‖_{L^{\infty} (Q_{R / 2})}^{s} ⩽ C ⨏_{Q_{2 R}} | \nabla u |^{s} d x d t + C ⨏_{Q_{2 R}} | F |^{s} d x d t, \\ (2.11) & ⨏_{Q_{R}} {| \nabla (u - v) |}^{s} d x d t ⩽ C {({[A]}_{R_{0}})}^{s} ⨏_{Q_{2 R}} | \nabla u |^{s} d x d t + C ⨏_{Q_{2 R}} | F |^{s} d x d t, \end{array}$ where C depends only on N, Λ, s.
Proof.
By standard interior regularity and inequality (2.6) in Lemma 2.2 and (2.9) in Lemma 2.3, we have $\begin{array}{l} ‖ \nabla v ‖_{L^{\infty} (Q_{R / 2})} & ⩽ C {(⨏_{Q_{R}} | \nabla v |^{2} d x d t)}^{1 / 2} ⩽ C {(⨏_{Q_{R}} | \nabla w |^{2} d x d t)}^{1 / 2} ⩽ C ⨏_{Q_{2 R}} | \nabla w | d x d t . \end{array}$ Thus, from this and inequality (2.4), it follows (2.10). On the other hand, applying (2.8) in Lemma 2.3 yields $\begin{array}{l} ⨏_{Q_{R}} {| \nabla (u - v) |}^{s} d x d t ⩽ C ⨏_{Q_{R}} {| \nabla (u - w) |}^{s} d x d t + C {({[A]}_{R})}^{s} ⨏_{Q_{2 R}} | \nabla w |^{s} d x d t . \end{array}$ Combining with (2.4), we get (2.11). The proof is complete. □

2.2. Boundary estimates

In this subsection, we handle with the corresponding estimates near the boundary. Throughout this subsection, Ω is a $(δ / 4, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ / 4$ for $δ < δ_{0}$ . Let $x_{0} \in \partial Ω$ be a boundary point and $0 < R < R_{0}$ and $t_{0} \in (0, T)$ . Since, for any $η > 0$ , $B_{\frac{1}{8}} ((0, \dots, \frac{1}{8} - ε)) \cap B_{1} (0)$ is $(η, η_{0})$ -Lip domain for some $ε > 0$ and $η_{0} > 0$ , there are a ball B of radius $R / 8$ and $ε_{1}, ε_{2} > 0$ depending only on N such that $B_{ε_{1} R} (x_{0}) \subset B \subset B_{R} (x_{0})$ and $B \cap Ω$ is $(δ, ε_{2} R) -$ Lip domain.

We set ${\tilde{Ω}}_{R / 8} = {\tilde{Ω}}_{R / 8} (x_{0}, t_{0}) = (Ω \cap B) \times (t_{0} - {(R / 8)}^{2}, t_{0})$ . Since $B \cap Ω$ is $(δ_{0}, ε_{2} R) -$ Lip domain and ${[A]}_{ε_{2} R} ⩽ δ_{0}$ , we apply Theorem 2.1 to $Ω_{T} = {\tilde{Ω}}_{R / 8}$ , $G = F$ and $q = s$ to obtain that there exists a unique very weak solution W to $\begin{matrix} (2.12) & \{\begin{matrix} W_{t} - div (A (x, t) \nabla W) = div (F) & in {\tilde{Ω}}_{R / 8}, \\ W = 0 & on \partial_{p} {\tilde{Ω}}_{R / 8}, \end{matrix} \end{matrix}$ satisfying $\begin{matrix} (2.13) & ‖ \nabla W ‖_{L^{s} ({\tilde{Ω}}_{R / 8})} ⩽ C ‖ F ‖_{L^{s} ({\tilde{Ω}}_{R / 8})}, \end{matrix}$ where C depends only on N, Λ, s. Note that the constant $(diam (Ω) + T^{1 / 2}) / R_{0}$ in Theorem 2.1 equals $\frac{3}{8 ε_{2}}$ in this case. In what follows we extend F by zero to ${(Ω \times (- \infty, T))}^{c}$ , and W by zero to $R^{N + 1} ∖ {\tilde{Ω}}_{R / 8}$ . We now set $w = u - W$ , and it is easy to check that w is a weak solution of $\begin{matrix} (2.14) & \{\begin{matrix} w_{t} - div (A (x, t) \nabla w) = 0 & in {\tilde{Ω}}_{R / 8}, \\ w = u & on \partial_{p} {\tilde{Ω}}_{R / 8} . \end{matrix} \end{matrix}$

Lemma 2.5.
There exists a constant $C = C (N, Λ) > 0$ such that the following estimate $\begin{matrix} (2.15) & {(⨏_{Q_{ρ / 2} (y, s)} | \nabla w |^{2} d x d t)}^{\frac{1}{2}} ⩽ C ⨏_{Q_{3 ρ} (y, s)} | \nabla w | d x d t \end{matrix}$ holds for all $Q_{3 ρ} (z, s) \subset B \times (t_{0} - {(R / 8)}^{2}, t_{0})$ .

The above lemma was proved in [11, Theorem 7.5]. Now we set $ρ = ε_{1} R (1 - δ) / 8$ so that $0 < ρ / (1 - δ) < ε_{1} R_{0} / 8$ . By the definition of the Lipschitz domains and $B_{ε_{1} R} (x_{0}) \subset B$ , there exists a coordinate system ${y_{1}, y_{2}, \dots, y_{N}}$ with the origin $0 \in Ω$ such that in the coordinate system $x_{0} = (0, \dots, 0, - \frac{ρ δ}{4 (1 - δ)})$ and $B_{ρ} (0) \subset B$ , one has $\begin{matrix} B_{ρ}^{+} (0) \subset Ω \cap B_{ρ} (0) \subset B_{ρ} (0) \cap {y = (y_{1}, y_{2}, \dots, y_{N}) : y_{N} > - \frac{ρ δ}{2 (1 - δ)}} . \end{matrix}$ Since $δ < 1 / 4$ , we have $\begin{matrix} (2.16) & B_{ρ}^{+} (0) \subset Ω \cap B_{ρ} (0) \subset B_{ρ} (0) \cap {y = (y_{1}, y_{2}, \dots, y_{N}) : y_{N} > - ρ δ}, \end{matrix}$ where $B_{ρ}^{+} (0) : = B_{ρ} (0) \cap {y = (y_{1}, y_{2}, \dots, y_{N}) : y_{N} > 0}$ .

Furthermore, we consider the unique solution $\begin{matrix} v \in L^{2} (t_{0} - ρ^{2}, t_{0}; W^{1, 2} (Ω \cap B_{ρ} (0))) \end{matrix}$ to the following problem $\begin{matrix} (2.17) & \{\begin{matrix} v_{t} - div ({\overline{A}}_{B_{ρ} (0)} (t) \nabla v) = 0 & in {\tilde{Ω}}_{ρ} (0), \\ v = w & on \partial_{p} {\tilde{Ω}}_{ρ} (0), \end{matrix} \end{matrix}$ where ${\tilde{Ω}}_{ρ} (0) = (Ω \cap B_{ρ} (0)) \times (t_{0} - ρ^{2}, t_{0})$ . We put $v = w$ outside ${\tilde{Ω}}_{ρ} (0)$ . As in Lemma 2.3 (see [10, Lemma 2.8]), we obtain the following result.
Lemma 2.6.
There exists a positive constant $C = C (N, Λ)$ such that $\begin{array}{l} (2.18) & ⨏_{Q_{ρ} (0, t_{0})} {| \nabla (w - v) |}^{2} d x d t ⩽ C {({[A]}_{R})}^{2} ⨏_{Q_{ρ} (0, t_{0})} | \nabla w |^{2} d x d t, \\ (2.19) & C^{- 1} \int_{Q_{ρ} (0, t_{0})} | \nabla v |^{2} d x d t ⩽ \int_{Q_{ρ} (0, t_{0})} | \nabla w |^{2} d x d t ⩽ C \int_{Q_{ρ} (0, t_{0})} | \nabla v |^{2} d x d t . \end{array}$

We can see that if the boundary of Ω is irregular enough, then the $L^{\infty}$ -norm of $\nabla v$ up to $\partial Ω \cap B_{ρ} (0) \times (t_{0} - ρ^{2}, t_{0})$ may not exist. However, we have the following lemma obtained in [11, Lemma 7.12]. Lemma 2.7.
For any $ε > 0$ , there exists a $δ_{1} = δ_{1} (N, Λ, ε) \in (0, δ_{0})$ such that if $δ \in (0, δ_{1})$ , there exists a function $V \in C (t_{0} - ρ^{2}, t_{0}; L^{2} (B_{ρ}^{+} (0))) \cap L^{2} (t_{0} - ρ^{2}, t_{0}; W^{1, 2} (B_{ρ}^{+} (0)))$ satisfying $\begin{array}{l} ‖ \nabla V ‖_{L^{\infty} (Q_{ρ / 4} (0, t_{0}))}^{2} ⩽ C ⨏_{Q_{ρ} (0, t_{0})} | \nabla v |^{2} d x d t, \\ ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (v - V) |}^{2} d x d t ⩽ ε^{2} ⨏_{Q_{ρ} (0, t_{0})} | \nabla v |^{2} d x d t, \end{array}$ for some $C = C (N, Λ) > 0$ .
Proposition 2.8.
For any $ε > 0$ there exists a $δ_{1} = δ_{1} (N, Λ, s, q_{0}, ε) \in (0, δ_{0})$ such that the following holds. If Ω is a $(δ / 4, R_{0})$ -Lip domain with $δ \in (0, δ_{1})$ , there is a function $V \in L^{2} (t_{0} - {(R / 9)}^{2}, t_{0}; W^{1, 2} (B_{R / 9} (x_{0}))) \cap L^{\infty} (t_{0} - {(R / 9)}^{2}, t_{0}; W^{1, \infty} (B_{R / 9} (x_{0})))$ such that $\begin{array}{l} (2.20) & ‖ \nabla V ‖_{L^{\infty} (Q_{ε_{1} R / 500})}^{s} ⩽ C ⨏_{Q_{R}} | \nabla u |^{s} d x d t + C ⨏_{Q_{R}} | F |^{s} d x d t, \\ (2.21) & ⨏_{Q_{ε_{1} R / 500}} {| \nabla (u - V) |}^{s} d x d t ⩽ C (ε^{s} + {({[A]}_{R_{0}})}^{s}) ⨏_{Q_{R}} | \nabla u |^{s} d x d t + C ⨏_{Q_{R}} | F |^{s} d x d t, \end{array}$ for some $C = C (N, Λ, s) > 0$ . Here $Q_{ρ} = Q_{ρ} (x_{0}, t_{0})$ for all $ρ > 0$ .
Proof.
We can assume that $δ \in (0, 1 / 100)$ . So $\begin{matrix} (2.22) & Q_{ε_{1} R / 500} \subset Q_{ρ / 8} (0, t_{0}) \subset Q_{6 ρ} (0, t_{0}) \subset Q_{ε_{1} R} \subset Q_{R} \end{matrix}$ By Lemma 2.7 for any $ε > 0$ , we can find a positive $δ = δ (N, Λ, s, q_{0}, ε) < 1 / 100$ such that there is a function $V \in L^{2} (t_{0} - ρ^{2}, t_{0}; W^{1, 2} (B_{ρ} (0))) \cap L^{\infty} (t_{0} - ρ^{2}, t_{0}; W^{1, \infty} (B_{ρ} (0)))$ satisfying $\begin{array}{l} ‖ \nabla V ‖_{L^{\infty} (Q_{ρ / 4} (0, t_{0}))}^{2} ⩽ C ⨏_{Q_{ρ} (0, t_{0})} | \nabla v |^{2} d x d t, \\ ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (v - V) |}^{2} ⩽ ε^{2} ⨏_{Q_{ρ} (0, t_{0})} | \nabla v |^{2} d x d t . \end{array}$ Then, by (2.19) in Lemma 2.6 and (2.15) in Lemma 2.5 and (2.22), we get $\begin{array}{l} (2.23) & ‖ \nabla V ‖_{L^{\infty} (Q_{ε_{1} R / 500})}^{s} ⩽ C {(⨏_{Q_{ρ} (0, t_{0})} | \nabla w |^{2} d x d t)}^{\frac{s}{2}} ⩽ C ⨏_{Q_{ε_{1} R}} | \nabla w |^{s} d x d t, \\ (2.24) & ⨏_{Q_{ε_{1} R / 500}} {| \nabla (v - V) |}^{s} d x d t ⩽ C ε^{s} ⨏_{Q_{ε_{1} R}} | \nabla w |^{s} d x d t . \end{array}$ Therefore, from (2.13) and (2.23), we get (2.20).

Next we prove (2.21). Since (2.22), $\begin{array}{l} ⨏_{Q_{ε_{1} R / 500}} {| \nabla (u - V) |}^{s} d x d t ⩽ & C ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (u - V) |}^{s} d x d t \\ ⩽ & C ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (u - w) |}^{s} d x d t + C ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (w - v) |}^{s} d x d t \\ + C ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (v - V) |}^{s} d x d t . \end{array}$ Using (2.13) and (2.18), (2.19) in Lemma 2.6 and (2.24), we find that $\begin{array}{l} ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (u - w) |}^{s} d x d t ⩽ C ⨏_{Q_{R}} | F |^{s} d x d t, \\ \begin{matrix} ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (v - w) |}^{s} d x d t & ⩽ C {({[A]}_{R_{0}})}^{s} ⨏_{Q_{ε_{1} R}} | \nabla w |^{s} d x d t \\ ⩽ C {({[A]}_{R_{0}})}^{s} (⨏_{Q_{R}} | \nabla u |^{s} d x d t + ⨏_{Q_{R}} | F |^{s} d x d t), \end{matrix} \end{array}$ and $\begin{array}{l} ⨏_{Q_{ρ / 8} (0, t_{0})} {| \nabla (v - V) |}^{s} d x d t ⩽ C ε^{s} (⨏_{Q_{R}} | \nabla u |^{s} d x d t + ⨏_{Q_{R}} | F |^{s} d x d t) . \end{array}$ Hence, we derive (2.21). This completes the proof. □

3. Global integral gradient bounds for parabolic equations

The following good-λ type estimate will be essential for our global estimates later.

Theorem 3.1.

Let $w \in A_{\infty}$ , $F \in (L^{s} \cap L^{2}) (Ω_{T}, R^{N})$ and $s > 1$ . For any $ε > 0$ , $R_{0} > 0$ one finds $δ_{1} = δ_{1} (N, Λ, s, ε, {[w]}_{A_{\infty}}) \in (0, 1 / 2)$ and $δ_{2} = δ_{2} (N, Λ, s, ε, {[w]}_{A_{\infty}}, T_{0} / R_{0}) \in (0, 1)$ and $Λ_{0} = Λ_{0} (N, Λ, s) > 0$ such that if Ω is a $(δ_{1}, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ_{1}$ , then there exists a unique solution $u \in L^{s} (0, T; W_{0}^{1, s} (Ω)) \cap L^{2} (0, T; W_{0}^{1, 2} (Ω))$ to the problem ( 1.1 ) satisfying $\begin{matrix} (3.1) & w ({M (| \nabla u |^{s}) > Λ_{0} λ, M (| F |^{s}) ⩽ δ_{2} λ} \cap Ω_{T}) ⩽ C ε w ({M (| \nabla u |^{s}) > λ} \cap Ω_{T}) \end{matrix}$ for all $λ > 0$ , where the constant C depends only on N, Λ, s, $T_{0} / R_{0}$ , ${[w]}_{A_{\infty}}$ .

In order to prove above estimate, we will utilize L. Caffarelli and I. Peral’s technique in [4] which is the following technical lemma. Its proof is a consequence of the Lebesgue differentiation Theorem and the standard Vitali covering lemma which can be found in [2,8] with some modifications to fit the setting here.

Lemma 3.2.

Let Ω be a $(δ, R_{0})$ -Reifenberg flat domain with $δ < 1 / 4$ and let w be an $A_{\infty}$ weight. Suppose that the sequence of balls ${B_{r} (y_{i})}_{i = 1}^{L}$ with centers $y_{i} \in \overline{Ω}$ and radius $r ⩽ R_{0} / 4$ covers Ω. Set $s_{i} = T - i r^{2} / 2$ for all $i = 0, 1, \dots, [\frac{2 T}{r^{2}}]$ . Let $E \subset F \subset Ω_{T}$ be measurable sets for which there exists $0 < ε < 1$ such that $w (E) < ε w ({\tilde{Q}}_{r} (y_{i}, s_{j}))$ for all $i = 1, \dots, L$ , $j = 0, 1, \dots, [\frac{2 T}{r^{2}}]$ ; and for all $(x, t) \in Ω_{T}$ , $ρ \in (0, 2 r]$ , we have ${\tilde{Q}}_{ρ} (x, t) \cap Ω_{T} \subset F$ if $w (E \cap {\tilde{Q}}_{ρ} (x, t)) ⩾ ε w ({\tilde{Q}}_{ρ} (x, t))$ . Then $w (E) ⩽ ε B w (F)$ for a constant B depending only on N and ${[w]}_{A_{\infty}}$ .

Proof of Theorem 3.1.

By Theorem 2.1, we find $δ_{0} = δ_{0} (N, Λ, s)$ , then there exists a unique solution $u \in L^{s} (0, T; W_{0}^{1, s} (Ω))$ to problem (1.1) satisfying $\begin{array}{l} (3.2) & \int_{Ω_{T}} | \nabla u |^{s} d x d t ⩽ C \int_{Ω_{T}} | F |^{s} d x d t \end{array}$ provided that Ω is a $(δ, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ$ for $δ < δ_{0} = δ_{0} (N, Λ, s)$ and $R_{0} > 0$ . Let $ε \in (0, 1)$ . Set $\begin{matrix} E_{λ, δ_{2}} = {M (| \nabla u |^{s}) > Λ_{0} λ, M (| F |^{s}) ⩽ δ_{2} λ} \cap Ω_{T}, F_{λ} = {M (| \nabla u |^{s}) > λ} \cap Ω_{T} \end{matrix}$ for $δ_{2} \in (0, 1)$ , $Λ_{0} > 0$ and $λ > 0$ . Let ${y_{i}}_{i = 1}^{L} \subset Ω$ and a ball $B_{0}$ with radius $2 T_{0}$ such that $\begin{matrix} Ω \subset ⋃_{i = 1}^{L} B_{r_{0}} (y_{i}) \subset B_{0}, r_{0} = min {R_{0} / 1000, T_{0}} . \end{matrix}$ Let $s_{j} = T - j r_{0}^{2} / 2$ for all $j = 0, 1, \dots, [\frac{2 T}{r_{0}^{2}}]$ and $Q_{2 T_{0}} = B_{0} \times (T - 4 T_{0}^{2}, T)$ . Thus, $\begin{matrix} Ω_{T} \subset ⋃_{i, j} Q_{r_{0}} (y_{i}, s_{j}) \subset Q_{2 T_{0}} . \end{matrix}$ We verify that $\begin{matrix} (3.3) & w (E_{λ, δ_{2}}) ⩽ ε w ({\tilde{Q}}_{r_{0}} (y_{i}, s_{j})) \forall λ > 0, \end{matrix}$ for some $δ_{2} > 0$ small enough depending on N, s, ϵ, ${[w]}_{A_{\infty}}$ , $T_{0} / R_{0}$ .

Indeed, we can assume that $E_{λ, δ_{2}} \neq \emptyset$ , it follows $\int_{Ω_{T}} | F |^{s} d x d t ⩽ C | Q_{2 T_{0}} | δ_{2} λ$ . Since $M$ is a bounded operator from $L^{1} (R^{N + 1})$ into $L^{1, \infty} (R^{N + 1})$ and (3.2), we get $\begin{array}{l} | E_{λ, δ_{2}} | ⩽ \frac{C}{Λ λ} \int_{Ω_{T}} | \nabla u |^{s} d x d t ⩽ \frac{C}{Λ λ} \int_{Ω_{T}} | F |^{s} d x d t ⩽ C δ_{2} | Q_{2 T_{0}} |, \end{array}$ which implies $\begin{array}{l} w (E_{λ, δ_{2}}) ⩽ c {(\frac{| E_{λ, δ_{2}} |}{| Q_{2 T_{0}} |})}^{ν} w (Q_{2 T_{0}}) ⩽ C δ_{2}^{ν} w (Q_{2 T_{0}}), \end{array}$ where $(c, ν) = {[w]}_{A_{\infty}}$ . It is well-known that (see, e.g [7]) there exist $c_{1} = c_{1} (N, c, ν)$ and $ν_{1} = ν_{1} (N, c, ν)$ such that $\begin{matrix} \frac{w (Q_{2 T_{0}})}{w ({\tilde{Q}}_{r_{0}} (y_{i}, s_{j}))} ⩽ c_{1} {(\frac{| Q_{2 T_{0}} |}{| {\tilde{Q}}_{r_{0}} (y_{i}, s_{j}) |})}^{ν_{1}} \forall i, j . \end{matrix}$ Therefore, $\begin{array}{l} w (E_{λ, δ_{2}}) ⩽ C δ_{2}^{ν} c_{1} {(\frac{| Q_{2 T_{0}} |}{| {\tilde{Q}}_{r_{0}} (y_{i}, s_{j}) |})}^{ν_{1}} w ({\tilde{Q}}_{r_{0}} (y_{i}, s_{j})) < ε w ({\tilde{Q}}_{r_{0}} (y_{i}, s_{j})) \forall i, j, \end{array}$ for $δ_{2}$ small enough depending on N, s, ϵ, ${[w]}_{A_{\infty}}$ , $T_{0} / R_{0}$ . Hence, (3.3) follows.

Next we verify that for all $(x, t) \in Ω_{T}$ , $r \in (0, 2 r_{0}]$ and $λ > 0$ , we have ${\tilde{Q}}_{r} (x, t) \cap Ω_{T} \subset F_{λ}$ provided $\begin{matrix} w (E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} (x, t)) ⩾ ε w ({\tilde{Q}}_{r} (x, t)), \end{matrix}$ for some $δ_{2}$ small enough depending on N, s, ϵ, ${[w]}_{A_{\infty}}$ , $T_{0} / R_{0}$ . Indeed, take $(x, t) \in Ω_{T}$ and $0 < r ⩽ 2 r_{0}$ , we set ${\tilde{Q}}_{ρ} = {\tilde{Q}}_{ρ} (x, t) \forall ρ > 0$ . Now assume that ${\tilde{Q}}_{r} \cap Ω_{T} \cap F_{λ}^{c} \neq \emptyset$ and $E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} \neq \emptyset$ i.e, there exist $(x_{1}, t_{1}), (x_{2}, t_{2}) \in {\tilde{Q}}_{r} \cap Ω_{T}$ such that $M (| \nabla u |^{s}) (x_{1}, t_{1}) ⩽ λ$ and $M (| F |^{s}) (x_{2}, t_{2}) ⩽ δ_{2} λ$ . We need to prove that $\begin{matrix} (3.4) & w (E_{λ, δ_{2}} \cap {\tilde{Q}}_{r}) < ε w ({\tilde{Q}}_{r}) . \end{matrix}$ Using $M (| \nabla u |^{2}) (x_{1}, t_{1}) ⩽ λ$ , we can see that $\begin{matrix} M (| \nabla u |^{s}) (y, t^{'}) ⩽ max {M (χ_{{\tilde{Q}}_{2 r}} | \nabla u |^{s}) (y, t^{'}), 3^{N + 2} λ} \forall (y, t^{'}) \in {\tilde{Q}}_{r} . \end{matrix}$ Therefore, for all $λ > 0$ and $Λ_{0} ⩾ 3^{N + 2}$ , $\begin{array}{rcl} (3.5) & E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} = {M (χ_{{\tilde{Q}}_{2 r}} | \nabla u |^{s}) > Λ_{0} λ, M (| F |^{s}) ⩽ δ_{2} λ} \cap Ω_{T} \cap {\tilde{Q}}_{r} . \end{array}$ In particular, $E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} = \emptyset$ if ${\overline{B}}_{8 r} (x) \subset \subset R^{N} ∖ Ω$ . Thus, it is enough to consider the case $B_{8 r} (x) \subset \subset Ω$ and the case $B_{8 r} (x) \cap Ω \neq \emptyset$ .

Firstly, we assume $B_{8 r} (x) \subset \subset Ω$ . Let v be in Proposition 2.4 with $Q_{2 R} = Q_{8 r} (x, t_{0})$ and $t_{0} = min {t + 2 r^{2}, T}$ . We have $\begin{array}{l} (3.6) & ‖ \nabla v ‖_{L^{\infty} (Q_{2 r} (x, t_{0}))}^{s} ⩽ C ⨏_{Q_{8 r} (x, t_{0})} | \nabla u |^{s} d x d t + C ⨏_{Q_{8 r} (x, t_{0})} | F |^{s} d x d t, \\ ⨏_{Q_{4 r} (x, t_{0})} {| \nabla (u - v) |}^{s} d x d t ⩽ C ⨏_{Q_{8 r} (x, t_{0})} | F |^{s} d x d t + C {({[A]}_{R_{0}})}^{s} ⨏_{Q_{8 r} (x, t_{0})} | \nabla u |^{s} d x d t . \end{array}$ Here constants C in above two depend only N, Λ, s.

Thanks to $M (| \nabla u |^{s}) (x_{1}, t_{1}) ⩽ λ$ and $M (| F |^{s}) (x_{2}, t_{2}) ⩽ δ_{2} λ$ with $(x_{1}, t_{1}), (x_{2}, t_{2}) \in Q_{r} (x, t)$ , we find $Q_{8 r} (x, t_{0}) \subset {\tilde{Q}}_{17 r} (x_{1}, t_{1}), {\tilde{Q}}_{17 r} (x_{2}, t_{2})$ and $\begin{matrix} (3.7) & ‖ \nabla v ‖_{L^{\infty} (Q_{2 r} (x, t_{0}))}^{s} ⩽ C ⨏_{{\tilde{Q}}_{17 r} (x_{1}, t_{1})} | \nabla u |^{s} d x d t + C ⨏_{{\tilde{Q}}_{17 r} (x_{2}, t_{2})} | F |^{s} d x d t ⩽ C λ, \end{matrix}$ and $\begin{matrix} (3.8) & ⨏_{Q_{4 r} (x, t_{0})} {| \nabla (u - v) |}^{2} d x d t ⩽ C δ_{2} λ + C {({[A]}_{R_{0}})}^{s} λ ⩽ C (δ_{2} + δ_{1}^{s}) λ . \end{matrix}$ Here we used ${[A]}_{R_{0}} ⩽ δ_{1}$ in the last inequality.

In view of (3.7), we imply that for $Λ_{0} ⩾ max {3^{N + 2}, 2 C}$ and C is the constant in (3.7) $\begin{array}{l} | {M (χ_{{\tilde{Q}}_{2 r}} | \nabla v |^{s}) > Λ_{0} λ / 4} \cap {\tilde{Q}}_{r} | = 0 . \end{array}$ It follows that $\begin{array}{l} | E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} | & ⩽ | {M (χ_{{\tilde{Q}}_{2 r}} {| \nabla (u - v) |}^{s}) > Λ_{0} λ / 4} \cap {\tilde{Q}}_{r} | . \end{array}$ Since $M$ is a bounded operator from $L^{1}$ into $L^{1, \infty}$ and (3.8), ${\tilde{Q}}_{2 r} \subset Q_{4 r} (x, t_{0})$ , we deduce $\begin{array}{l} | E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} | ⩽ \frac{C}{λ} \int_{{\tilde{Q}}_{2 r}} {| \nabla (u - v) |}^{s} d x d t ⩽ C (δ_{2} + δ_{1}^{s}) | {\tilde{Q}}_{r} | . \end{array}$ Thus, $\begin{array}{l} w (E_{λ, δ_{2}} \cap {\tilde{Q}}_{r}) ⩽ c {(\frac{| E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} |}{| {\tilde{Q}}_{r} |})}^{ν} w ({\tilde{Q}}_{r}) ⩽ C {(δ_{2} + δ_{1}^{2})}^{ν} w ({\tilde{Q}}_{r}) < ε w ({\tilde{Q}}_{r}), \end{array}$ where $δ_{2}, δ_{1} ⩽ δ (N, Λ, s, ε, {[w]}_{A_{\infty}})$ and $(c, ν) = {[w]}_{A_{\infty}}$ .

Secondly, we assume $B_{8 r} (x) \cap Ω \neq \emptyset$ . Let $x_{3} \in \partial Ω$ such that $| x_{3} - x | = dist (x, \partial Ω)$ . Set $t_{0} = min {t + 2 r^{2}, T}$ . We have $\begin{matrix} (3.9) & \begin{matrix} Q_{2 r} (x, t_{0}) & \subset Q_{10 r} (x_{3}, t_{0}) \subset Q_{5000 r / ε_{1}} (x_{3}, t_{0}) \\ \subset {\tilde{Q}}_{10^{4} r / ε_{1}} (x_{3}, t) \subset {\tilde{Q}}_{10^{5} r} (x, t) \subset {\tilde{Q}}_{10^{6} r} (x_{1}, t_{1}), \end{matrix} \end{matrix}$ and $\begin{matrix} (3.10) & Q_{5000 r / ε_{1}} (x_{3}, t_{0}) \subset {\tilde{Q}}_{10^{4} r / ε_{1}} (x_{3}, t) \subset {\tilde{Q}}_{10^{5} r} (x, t) \subset {\tilde{Q}}_{10^{6} r} (x_{2}, t_{2}) . \end{matrix}$ Applying Theorem 2.8 with $Q_{R} = Q_{5000 r / ε_{1}} (x_{3}, t_{0})$ and $ε = δ_{3} \in (0, 1)$ , there exists a constant $δ_{0}^{'} = δ_{0}^{'} (N, Λ, s, δ_{3}) \in (0, δ_{0})$ such that if Ω is a $(δ_{0}^{'}, R_{0})$ -Lip domain, then $\begin{matrix} ‖ \nabla V ‖_{L^{\infty} (Q_{10 r} (x_{3}, t_{0}))}^{s} ⩽ C ⨏_{Q_{5000 r / ε_{1}} (x_{3}, t_{0})} | \nabla u |^{s} + C ⨏_{Q_{5000 r / ε_{1}} (x_{3}, t_{0})} | F |^{s}, \end{matrix}$ and $\begin{array}{l} ⨏_{Q_{10 r} (x_{3}, t_{0})} {| \nabla (u - V) |}^{s} ⩽ C (δ_{3}^{s} + {[A]}_{R_{0}}^{s}) ⨏_{Q_{5000 r / ε_{1}} (x_{3}, t_{0})} | \nabla u |^{s} + C ⨏_{Q_{5000 r / ε_{1}} (x_{3}, t_{0})} | F |^{s} . \end{array}$ Since $M (| \nabla u |^{s}) (x_{1}, t_{1}) ⩽ λ$ , $M (| F |^{s}) (x_{2}, t_{2}) ⩽ δ_{2} λ$ and (3.9), (3.10), we get $\begin{array}{l} (3.11) & ‖ \nabla V ‖_{L^{\infty} (Q_{10 r} (x_{3}, t_{0}))}^{s} ⩽ C ⨏_{{\tilde{Q}}_{10^{6} r} (x_{1}, t_{1})} | \nabla u |^{s} d x d t + C ⨏_{{\tilde{Q}}_{10^{6} r} (x_{1}, t_{1})} | F |^{s} d x d t ⩽ C λ, \\ (3.12) & ⨏_{Q_{10 r} (x_{3}, t_{0})} {| \nabla (u - V) |}^{s} d x d t ⩽ C (δ_{3}^{s} + {({[A]}_{R_{0}})}^{s} + δ_{2}) λ ⩽ C (δ_{3}^{s} + δ_{1}^{s} + δ_{2}) λ . \end{array}$ Notice that we have used ${[A]}_{R_{0}} ⩽ δ_{1}$ in the last inequality.

We also obtain that for $Λ_{0} ⩾ max {3^{N + 2}, 4 C}$ and the constant C is in (3.11), $\begin{array}{l} | E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} | & ⩽ | {M (χ_{{\tilde{Q}}_{2 r}} {| \nabla (u - V) |}^{s}) > Λ_{0} λ / 4} \cap {\tilde{Q}}_{r} |, \end{array}$ where the constant $Λ_{0}$ depends only on N, Λ, s.

Thanks to (3.12), we obtain $\begin{matrix} | E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} | ⩽ \frac{C}{λ} \int_{{\tilde{Q}}_{2 r}} {| \nabla (u - V) |}^{s} d x d t ⩽ C (δ_{3}^{s} + δ_{1}^{s} + δ_{2}) | {\tilde{Q}}_{r} | . \end{matrix}$ Thus, $\begin{matrix} w (E_{λ, δ_{2}} \cap {\tilde{Q}}_{r}) ⩽ c {(\frac{| E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} |}{| {\tilde{Q}}_{r} |})}^{ν} w ({\tilde{Q}}_{r}) ⩽ C {(δ_{3}^{s} + δ_{1}^{s} + δ_{2})}^{ν} w ({\tilde{Q}}_{r}) < ε w ({\tilde{Q}}_{r}), \end{matrix}$ where $δ_{1}, δ_{2}, δ_{3} ⩽ δ^{'} (N, Λ, s, ε, {[w]}_{A_{\infty}})$ and $(c, ν) = {[w]}_{A_{\infty}}$ .

Therefore, for all $(x, t) \in Ω_{T}$ , $r \in (0, 2 r_{0}]$ and $λ > 0$ , if $w (E_{λ, δ_{2}} \cap {\tilde{Q}}_{r} (x, t)) ⩾ ε w ({\tilde{Q}}_{r} (x, t))$ , then ${\tilde{Q}}_{r} (x, t) \cap Ω_{T} \subset F_{λ}$ , where Ω is a $(δ_{1}, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ_{1}$ with $δ_{1} = δ_{1} (N, Λ, s, ε, {[w]}_{A_{\infty}}) \in (0, δ_{0})$ , $δ_{2} = δ_{2} (N, Λ, s, ε, {[w]}_{A_{\infty}}, T_{0} / R_{0}) \in (0, 1)$ . Hence, combining with (3.3), we can apply Lemma 3.2 to get the desired result. □

Proof of Theorem 1.1.

Thanks to the uniqueness of very weak solution to equation (1.1) in Theorem 2.1, it is enough to show that $\begin{matrix} (3.13) & | ‖ \nabla u | ‖_{L_{w}^{q, p} (Ω_{T})} ⩽ C ‖ F ‖_{L_{w}^{q, p} (Ω_{T})} . \end{matrix}$ under assumption $F \in L_{w}^{q, p} (Ω_{T}) \cap L^{2} (Ω_{T})$ . Thus, the solution u belongs to $L^{2} ((- \infty, T], W_{0}^{1, 2} (Ω))$ . Since $L_{w}^{q, p} (Ω_{T}) \subset L^{q_{0}} (Ω_{T})$ for some $q_{0} > 1$ . By Theorem 2.1, we find $δ_{0} = δ_{0} (N, Λ, q, p)$ , such that there exists a unique very weak solution $u \in L^{q_{0}} (0, T; W_{0}^{1, q_{0}} (Ω))$ to problem (1.1) satisfying $\begin{array}{l} \int_{Ω_{T}} | \nabla u |^{q_{0}} d x d t ⩽ C \int_{Ω_{T}} | F |^{q_{0}} d x d t \end{array}$ provided that Ω is a $(δ, R_{0})$ -Lip domain and ${[A]}_{R_{0}} ⩽ δ$ for $δ < δ_{0} = δ_{0} (N, Λ, q, p)$ .

By Theorem 3.1, for any $s \in (1, q_{0})$ , $ε > 0$ , $R_{0} > 0$ one finds $δ = δ (N, Λ, ε, s, {[w]}_{A_{\infty}}) \in (0, δ_{0})$ and $δ_{2} = δ_{2} (N, Λ, ε, s, {[w]}_{A_{\infty}}, T_{0} / R_{0}) \in (0, 1)$ and $Λ_{0} = Λ_{0} (N, Λ) ⩾ 1$ such that if Ω is a $(δ, R_{0})$ - Lip domain and ${[A]}_{R_{0}} ⩽ δ$ , then $\begin{matrix} (3.14) & w ({M (| \nabla u |^{s}) > Λ_{0} λ, M [| F |^{s}] ⩽ δ_{2} λ} \cap Ω_{T}) ⩽ C ε w ({M (| \nabla u |^{s}) > λ} \cap Ω_{T}), \end{matrix}$ for all $λ > 0$ , where the constant C depends only on N, Λ, s, $T_{0} / R_{0}$ , ${[w]}_{A_{\infty}}$ . Thus, for $q_{1}, p_{1} < \infty$ , $γ < \infty$ $\begin{array}{l} q_{1} \int_{0}^{γ} λ^{p_{1}} {(w ({M (| \nabla u |^{s}) > λ} \cap Ω_{T}))}^{p_{1} / q_{1}} \frac{d λ}{λ} \\ = q_{1} Λ_{0}^{p_{1}} \int_{0}^{\frac{γ}{Λ_{0}}} λ^{p_{1}} {(w ({M (| \nabla u |^{s}) > Λ_{0} λ} \cap Ω_{T}))}^{p_{1} / q_{1}} \frac{d λ}{λ} \\ ⩽ q_{1} Λ_{0}^{p_{1}} 2^{p_{1} / q_{1}} {(C ε)}^{p_{1} / q_{1}} \int_{0}^{γ} λ^{p_{1}} {(w ({M (| \nabla u |^{s}) > λ} \cap Ω_{T}))}^{p_{1} / q_{1}} \frac{d λ}{λ} \\ + Λ_{0}^{p_{1}} 2^{p_{1} / q_{1}} δ_{2}^{- p_{1}} {‖ M (| F |^{s}) ‖}_{L_{w}^{q_{1}, p_{1}} (Ω_{T})}^{p_{1}} . \end{array}$ We can choose $ε = ε (N, Λ, q_{1}, p_{1}, C) > 0$ such that $Λ_{0}^{p_{1}} 2^{p_{1} / q_{1}} {(C ε)}^{p_{1} / q_{1}} ⩽ 1 / 2$ , then we get $\begin{array}{l} \frac{q_{1}}{2} \int_{0}^{γ} λ^{p_{1}} {(w ({M (| \nabla u |^{s}) > λ} \cap Ω_{T}))}^{p_{1} / q_{1}} \frac{d λ}{λ} ⩽ Λ_{0}^{p_{1}} 2^{p_{1} / q_{1}} δ_{2}^{- p_{1}} {‖ M (| F |^{s}) ‖}_{L_{w}^{q_{1}, p_{1}} (Ω_{T})}^{p_{1}} . \end{array}$ Letting $γ \to \infty$ , one has $\begin{array}{l} (3.15) & {‖ M (| \nabla u |^{s}) ‖}_{L_{w}^{q_{1}, p_{1}} (Ω_{T})} ⩽ C {‖ M (| F |^{s}) ‖}_{L_{w}^{q_{1}, p_{1}} (Ω_{T})} . \end{array}$ and the above inequality is also true when $p_{1} = \infty$ .

Let $w \in A_{q}$ , there exists $q_{2} = q_{2} (N, q, {[w]}_{A_{q}}) \in (1, q)$ such that ${[w]}_{A_{q_{2}}} ⩽ C_{0} = C_{0} (N, q, {[w]}_{A_{q}})$ . Thus, $\begin{matrix} {‖ M (| F |^{\frac{q}{q_{2}}}) ‖}_{L_{w}^{q_{2}, p q_{2}} (Ω_{T})}^{\frac{q_{2}}{q}} ⩽ C ‖ F ‖_{L_{w}^{q, p} (Ω_{T})} . \end{matrix}$ Applying (3.15) to $s = q / q_{2}$ and $q_{1} = q_{2}$ , $p_{1} = p q_{2} / q$ , we have $\begin{array}{l} | ‖ \nabla u | ‖_{L_{w}^{q, p} (Ω_{T})} ⩽ {‖ M (| \nabla u |^{q / q_{2}}) ‖}_{L_{w}^{q_{2}, p q_{2}} (Ω_{T})}^{\frac{q_{2}}{q}} ⩽ C {‖ M (| F |^{q / q_{2}}) ‖}_{L_{w}^{q_{2}, p q_{2}} (Ω_{T})}^{\frac{q_{2}}{q}} ⩽ C ‖ F ‖_{L_{w}^{q, p} (Ω_{T})} . \end{array}$ This implies (3.13). The proof is complete. □

Footnotes

Acknowledgement

The research leading to the present results has received funding from research grant of Vietnam National University, HCM city, project number: B2019-18-01. The authors would like to thank the referee for his/her comments, which help to improve the presentation of this article.

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