Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaces

Dedicated to Giuseppe Rosario Mingione, on the occasion of his 50th birthday.

1.   Introduction

We study local regularity theory for weak solutions to the following parabolic equations of $p$-Laplacian type:

where $\Omega\subset \mathbb{R}^n$ ($n \geqslant 2$) is an open set, $T$ is a positive constant, $u = u(x, t)$ is a real valued function with $(x, t)\in \Omega\times (0, T] = \Omega_T$, $u_t$ is the partial derivative of $u$ with respect to the time variable $t$, and $Du\in \mathbb{R}^n$ is the gradient of $u$ with respect to the space variable $x$ (i.e., $Du = D_x u$). For given $p \in (1, \infty),$ we assume that $\mathbf{a}: \mathbb{R}^n\to \mathbb{R}^n$ satisfies the following $p$-growth and $p$-ellipticity conditions:

and

for every $\xi, \eta \in \mathbb{R}^n\setminus\{0\},$ and for some constants $\nu$ and $L$ with $0 < \nu \leqslant 1 \leqslant L.$ The prototype of $\mathbf a$ is

$L^q$-regularity theory with Calderón-Zygmund estimates for partial differential equations is originated from the classical result of Calderón and Zygmund ^[16] about the boundedness of linear operators including the Laplace operator. For the following $p$-Laplacian type equation

a fundamental $L^q$-regularity theory, which is also called (nonlinear) Calderón-Zygmund theory, is to show the following implication:

and obtain corresponding estimates, so-called Calderón-Zygmund estimates. In this regard, Iwaniec ^[26] first obtained Calderón-Zygmund estimates in the whole space $\mathbb{R}^n$ when $p \geqslant 2$, and DiBenedetto and Manfredi ^[21] extended this result to the corresponding system with $1 < p < \infty$. Thereafter, Caffarelli and Peral ^[15] considered general elliptic equations with $p$-growth, for instance, the stationary case of (1.1), applying a new approach by means of maximal functions and a covering argument obtained by Krylov and Safonov based on the Calderón-Zygmund decomposition. We further refer to e.g., ^{[1,11,14,17,28]} for $L^q$-regularity theory with corresponding Calderón-Zygmund estimates for elliptic problems.

Difficulty of the study on regularity theory for parabolic problems of $p$-Laplacian type is originated from the absence of the scaling invariant property: a constant multiple of a solution of the parabolic $p$-Laplace equation $u_t- \operatorname{div} (|Du|^{p-2}Du) = 0$ does not become a solution. It could be overcome by considering intrinsic parabolic cylinders depending on solutions, instead of the usual parabolic cylinders. This idea was introduced by DiBenedetto and Friedman in ^[19,20], see also the monograph ^[18], where Hölder regularity for parabolic $p$-Laplace systems had been established.

For the $L^q$-regularity theory, on the other hand, the approaches used in ^[26] and ^[15] are not directly applicable to the parabolic $p$-Laplacian type problems when $p\neq 2$, since the intrinsic geometry prevents the use of maximal functions. Finally, Acerbi and Mingione ^[2] established local Calderón-Zygmund estimates when $p > \frac{2n}{n+2}$ with a new approach, hence proved the implication (1.4) in the local sense. We also refer to the higher integrability result of Kinnunen and Lewis in ^[27], where the implication (1.4) is obtained when $q$ is sufficiently close to $1$. Note that the condition $p > \frac{2n}{n+2}$ is essential since there exists an unbounded weak solution to the parabolic $p$-Laplace system in this case, see ^[18]. It is worth pointing out that the approach in ^[2] does not employ maximal functions and the covering argument by Krylov and Safonov but a new covering argument used in ^[31] which is based on the Vitali covering lemma. It has led to the development of the Calderón- Zygmund theory for parabolic problems. For instance, we refer to ^[5,7,10] for global Calderón Zygmund theory in bounded domains, ^[6,33] for parabolic obstacle problems, ^[4,9] for parabolic problems with variable exponent, ^[25,32] for parabolic problems with growth and ^[35] for parabolic variational problems.

Research on Calderón-Zygmund estimates in general function spaces such as weighted Lebesgue spaces, Orlicz spaces, variable exponent Lebesgue spaces, Lorentz spaces have been actively conducted for the last decade, e.g., ^{[3,8,12,29,36]}. In particular, estimates in weighted Lebesgue spaces are crucial since these imply estimates in various function spaces by extrapolation argument, see [24, Section 5]. For parabolic problems with $p$-growth as in (1.1), Byun and Ryu ^[13] obtained global Calderón-Zygmund estimates in the weight Lebesgue spaces $L^q_w$ hence proved the following implication:

with $q > 1$ and the weight $w$ satisfying the following Muckenhoupt type condition:

where $\mathcal C$ is the set of all cylinders of the form $B_r(x_0)\times (t_1, t_2)\subset \mathbb{R}^n \times \mathbb{R}$. On the other hand, if $p = 2$, the same implication can be obtained for every usual parabolic $A_q$ weight $w$, that is, $w$ satisfies (1.6) with $\mathcal C$ the set of all parabolic cylinders of the form $B_r(x_0)\times (t_0-r^2, t_0+r^2)$. Therefore, there is a drastic change between the conditions of weights when $p = 2$ and $p\neq 2$.

In this paper, we introduce a new parabolic Muckenhoupt type condition depending on the intrinsic geometry concerned with the parabolic $p$-Laplacian setting, see Definition 2.1. We emphasize that our condition on weights depends on $p$, and is weaker than the one in ^[13] and exactly the same as the parabolic $A_q$ condition when $p = 2$. With this condition we prove the implication (1.5) in the local sense by obtaining corresponding Calderón-Zymund estimates.

Now, we state our main result. Notation and the definition of the $p$-intrinsic $A_q$ weight are introduced in next section. We say that $u\in C^0(0, T;L^2(\Omega))\cap L^p(0, T;W^{1, p}(\Omega))$ is a weak solution to (1.1) if

holds for every $\zeta\in C^\infty_0(\Omega_T)$.

Theorem 1.1. Let $p > \frac{2n}{n+2}$ and $u\in C^0(0, T;L^{2}(\Omega))\cap L^p(0, T;W^{1, p}(\Omega))$ be a weak solution to (1.1) with $F\in L^{p}(\Omega_T, \mathbb{R}^n)$. If $w$ is a $p$-intrinsic $A_q$ weight with $q > 1$ and $|F|^p\in L^q_{w, \mathrm{loc}}(\Omega_T)$, then $|Du|^p\in L^q_{w, \mathrm{loc}}(\Omega_T)$.

Furthermore, there exists $R_0 = R_0(n, \nu, L, p, q, [w]_q, Du, F) > 0$ such that for every $Q_{2r}\Subset \Omega_T$ with $2r < R_0$,

for some $c = c(n, \nu, L, p, q, [w]_q) > 0$, where

Remark 1.1. In the above theorem, $R_0$ will be chosen as in (3.1). Furthermore, when $p = 2$ we may put $R_0 = \infty$.

Remark 1.2. (Possible extensions) In this paper, we deal with only scalar problems without coefficients for simplicity. We can consider more general problems such as general non-autonomous parabolic equations with $p$-growth

where $\mathbf a(x, t, \xi)$ satisfies (1.2), (1.3) and a VMO condition (see ^[10]), and parabolic $p$-Laplace systems with coefficients

where $u : \Omega_T \to \mathbb{R}^N$ and $A(x, t): \mathbb{R}^{n+1}\to \mathbb{R}^{n^2N^2}$ satisfies a VMO condition (see ^[2]). Moreover, as in ^[13], we can also consider global Calderón-Zygmund estimates in Reifenberg flat domains.

The remaining part of the paper is organized as follows. In Section 2, we introduce notation, weights with their main assumption, and comparison and regularity estimates for corresponding homogeneous problems. In Section 3, we prove our main theorem, Theorem 1.1.

2.   Preliminaries

2.1.   Notation

Let $z_0 = (x_0, t_0)\in \mathbb{R}^n\times \mathbb{R}$ with $x_0 = (x_0^1, \dots, x_0^n)$, $r, \alpha, \lambda > 0$ and $1 < p < \infty$. We define an $\alpha$-parabolic cylinder by $Q_{r, \alpha}(z_0) = B_r(x_0)\times (t_0-\alpha r^2, t_0+\alpha r^2)$ and an $\alpha$-parabolic cube by $\tilde Q_{r, \alpha}(z_0) = C_r(x_0)\times (t_0-\alpha r^2, t_0+\alpha r^2)$, where $B_r(x_0): = \{x\in \mathbb{R}^n: |x-x_0| < r\}$ and $C_r(x_0): = \{x = (x^1, \dots, x^n)\in \mathbb{R}^n: \max\{|x^1-x_0^1|, \dots, |x^n-x_0^n|\} < r\}$. Note that $Q_r(z_0): = Q_{r, 1}(z_0)$ and $\tilde Q_r(z_0): = \tilde Q_{r, 1}(z_0)$ is the usual parabolic cylinder and cube, respectively, and we denote $\partial_{\mathrm{p}}Q_r(z_0): = (B_r(x_0)\times \{t = t_0-r^2\})\cup(\partial B_r(x_0) \times [t_0-r^2, t_0+r^2))$ Furthermore, when $\alpha = \lambda^{2-p}$ we write $Q^\lambda_r(z_0) = Q_{r, \lambda^{2-p}}(z_0) = B_r(x_0)\times (t_0-\lambda^{2-p} r^2, t_0+\lambda^{2-p} r^2)$ which is usually called a $p$-intrinsic parabolic cylinder since we will consider $\lambda$ related to the weak solution to (1.1).

For an integrable function $f: U\to \mathbb{R}^m$ with $U\subset \mathbb{R}^{n+1}$ and $0 < |U| < \infty$, we write $(f)_{U} = \mathit{{\rlap{-} \smallint }}_U f\, dz : = \frac{1}{|U|}\int_U f \, dz$, where $|U|$ is the Lebesgue measure of $U$ in $\mathbb{R}^{n+1}$.

2.2.   Weights

We say that $w: \mathbb{R}^{n+1}\to \mathbb{R}$ is a weight if it is nonnegative and locally integrable. For a weight $w$ and a bounded open set $U\subset \mathbb{R}^{n+1}$, we write

and define by weighted Lebesgue space $L^q_w(U)$, $1 \leqslant q < \infty$, the set of all measurable function $f$ on $U$ such that

We introduce the assumption of the weight $w$ in Theorem 1.1.

Definition 2.1. Let $p, q\in(1, \infty)$. We say that weight $w: \mathbb{R}^{n+1}\to \mathbb{R}$ is a $p$-intrinsic parabolic $A_q$ weight if it satisfies that

where

Here, the $\alpha$-parabolic cylinders $Q_{r, \alpha}(z_0)$ can be replaced by the $\alpha$-parabolic cubes $\tilde Q_{r, \alpha}(z_0)$.

Note that in the definition of the class $\mathcal C_p$, the range of $\alpha$ with respect to $p$ is following:

Hence, $\mathcal C_p$ contains all the parabolic cylinders $Q_r(z_0)$ and, in particular, $\mathcal C_2$ (i.e., $p = 2$) consists of only the parabolic cylinders. Moreover, since $r^{-\frac{n+2}{2}} \leqslant \rho^{-\frac{n+2}{2}}$ for $\rho\in(0, r]$, we have

From this fact, we can obtain the following properties for $p$-intrinsic parabolic $A_q$ weights, which are well known properties of the usual $A_q$ weights, see e.g., [23, Section 7.2]. Proofs are exactly the same as the ones in there with just replacing cubes and the dimension $n$ by the $\alpha$-parabolic cubes and $n+2$, respectively. Therefore, we omit their proofs.

Proposition 2.1. Let $p, q\in(1, \infty)$ and $w: \mathbb{R}^{n+1}\to \mathbb{R}$ be a $p$-intrinsic parabolic $A_q$ weight.

(1) For every $f\in L^q_w(Q)$ with $Q\in \mathcal C_p$,

(2) There exist $\gamma, c > 0$ depending on $n$, $q$ and $[w]_q$ such that

(3) There exist $\gamma_1 > 0$ and $c_1, c_2 \geqslant 1$ depending on $n$, $q$ and $[w]_q$ such that for every $Q\in \mathcal C_p$ and $E\subset Q$

(4) $w$ is a $p$-intrinsic parabolic $A_{q_1}$ weight for every $q_1 > q$. Moreover, $w$ is a $p$-intrinsic parabolic $A_{q'}$ weight for some $q'\in (1, q)$, where $q'$ and $[w]_{q'}$ depend on $n$, $q$ and $[w]_q$.

Example. On $\mathbb{R}^{n+1}$, the function $w(x, t) = \max\{|x|, \sqrt{|t|}\}^\gamma$ is a $p$-intrinsic $A_q$ weight for $p \geqslant 2$, $q\in(1, \infty)$ when $-n < \gamma < n(q-1)$. Indeed, we first write

We divide the $\alpha$-parabolic cylinders $Q_{r, \lambda^{2-p}}(z_0)$ in $\mathcal C_p$ for $z_0 = (x_0, t_0)\in \mathbb{R}^{n+1}$ into three cases:

(i) $\min\{|x_0|, \sqrt{|t_0|}\} \geqslant 3r$,

(ii) $|x_0| < 3r$ and $3\lambda^{\frac{2-p}{2}}r \leqslant \sqrt{|t_0|} < 3r$,

(iii) $|x_0| < 3r$ and $\sqrt{|t_0|} < 3\lambda^{\frac{2-p}{2}}r$.

Note that if $x\in B_r(x_0)$ and $|x_0| \geqslant 3r$,

and if $t_0-\lambda^{2-p}r^2 \leqslant t \leqslant t_0+\lambda^{2-p}r^2$ and $\sqrt{|t_0|} > 3 \lambda^{\frac{2-p}{2}}r$,

In Case (i), if $(x, t)\in Q_{r, \lambda^{2-p}}(z_0)$, we have $|x|\approx |x_0|$ and $|t|\approx |t_0|$, hence

In Case (ii), if $(x, t)\in Q_{r, \lambda^{2-p}}(z_0)$, then $|x| < 4r$ and $|t|\approx |t_0| < 9r^2$, and hence

where we have used $-n < \gamma < n (q-1)$.

In Case (iii), if $(x, t)\in Q_{r, \lambda^{2-p}}(z_0)$, then $|x| \leqslant 4r$ and $|t| \leqslant 10 \lambda^{2-p}r^2$, and hence, by the similar computation as in Case (ii), we have

Finally, using the facts that $-n < \gamma < n (q-1)$ and $\lambda \geqslant 1$,

2.3.   Comparison and Lipschitz regularity of homogeneous problems

We consider the following homogeneous problem in simple parabolic cylinder $Q_{2} = Q_{2}(0)$:

where $u \in C^0(-2^2, 2^2;L^{2}(B_2))\cap L^p(-2^2, 2^2;W^{1, p}(B_2))$ is a weak solution to (1.1) with replacing $\Omega_T$ by $Q_2$. For the existence and the uniqueness of the weak solution $h\in C^0(-2^2, 2^2;L^{2}(B_2))\cap L^p(-2^2, 2^2;$ $W^{1, p}(B_2))$ to the above equation, we refer to e.g., [34, Section III.4]. Then, we obtain the following regularity estimates for $h$ and comparison estimate between $u$ and $h$.

Lemma 2.1. Let $u$ be a weak solution to (1.1) in $Q_{2}$ with

for some $\delta\in(0, 1)$, and let $h$ be the weak solution to (2.4). Then

for some $c, c_{\mathrm{Lip}} \geqslant 1$ depending on $n, \nu, L$ and $p$, where $d \geqslant 1$ is from (1.8).

Moreover, for any $\varepsilon\in(0, 1)$, there exists small $\delta = \delta(n, \nu, L, p, \varepsilon)\in(0, 1)$ such that

Proof. In view of [18, Section VIII.5], we have the first inequality in (2.6). We note that the Lipschitz regularity estimates in ^[18] are obtained for the parabolic $p$-Laplace systems. However, the same argument can apply to equations of $p$-Laplacian type such as (1.1) with the nonlinearity $\mathbf a$ satisfying (1.2) and (1.3).

Regarding (2.7) and the second inequality in (2.6), similar comparison estimates can be found in numerous papers, see e.g., ^[2,5,10]. But, we shall prove them in details for completeness.

We take $\zeta = u-h$ as a test function in (1.1) and (2.4) to obtain

and

We notice that $u$ and $h$ are not differentiable for $t$. However, by considering their Steklov averages (see e.g., [18, Section I.3] and ^[6]), we may assume that they are differentiable for $t$. Then we have

Note that

We remark that the condition (1.3) implies the monotonicity condition:

for every $\xi, \eta \in \mathbb{R}^n\setminus\{0\}$. Then we see

Therefore, by the above estimates, Young's inequality and the second inequality in (2.5), we have that for any $\kappa_1\in(0, 1)$,

If $p \geqslant 2$, since $|Du-Dh|^p \leqslant (|Du|^2 + |Dh|^2)^{\frac{p-2}{2}} |Du-Dh|^2$, by taking sufficiently small $\kappa_1 = \kappa_1(n, \nu, L, p) > 0$ we have

The second inequality in (2.6) follows from the first inequality in (2.5) together with $\delta \leqslant 1$. Moreover, by choosing small $\delta$ depending on $\varepsilon$, we get (2.7).

If $\frac{2n}{n+2} < p < 2$, on the other hand, applying Young's inequality, we have that for any $\kappa_2\in(0, 1)$,

Hence by choosing $\kappa_1$ sufficiently small depending on $\kappa_2$ we have

We first note that

Then by choosing $\kappa_2$ sufficiently small and using the first inequality in (2.5) and $\delta \leqslant 1$ we have the second inequality in (2.6). Finally, applying the second inequalities of (2.5) and (2.6) to (2.8), we have

Finally, choosing $\kappa_2$ and $\delta$ sufficiently small depending on $\varepsilon$ we get (2.7).

3.   Proof of Theorem 1.1

Now we start with the proof of the main theorem, Theorem 1.1. As we mentioned in the introduction, we follow the approach introduced in ^[2], see also ^[13] for the case of the weighted Lebesgue space. We divide the proof into five steps.

Step 1. (Setting and stopping time argument)

Let $\delta\in (0, 1)$, which will be determined as a small constant depending only on $n$, $\nu$, $L$, $p$, $q$ and $[w]_q$ in below (3.17). Then there exists $R_0 > 0$ satisfying that

We fix any $Q_{2r} = Q_{2r}(z_0)\Subset \Omega_T$ with $2r < R_0$. For simplicity, we write $Q_\rho = Q_\rho(z_0)$, $\rho\in(0, 2r]$. In addition, for $\rho > 0$ and $\lambda > 0$, we define the super level set

and

where $d \geqslant 1$ is from (1.8).

Let $r \leqslant r_1 < r_2 \leqslant 2r$ and consider any $\lambda$ satisfying the following:

We notice that $Q^{\lambda}_{\rho}(\tilde{z}) \subset Q_{r_2}\subset Q_{2r}$ for any $\tilde{z} = (\tilde{x}, \tilde{t}) \in E(r_1, \lambda)$ and all $\rho < \rho_0$ where

Then we obtain the following Vitali type covering result for the super-level set $E(r_1, \lambda)$.

Lemma 3.1. For each $r \leqslant r_1 < r_2 \leqslant 2r$ and $\lambda \geqslant B\lambda_0$, there exist $z_i\in E(r_1, \lambda)$ and $\rho_i\in \left(0, \frac{\rho_0}{10} \right)$, $i = 1, 2, 3, \cdots$, such that the intrinsic parabolic cylinders $Q^\lambda_{\rho_i}(z_i)$ are mutually disjoint,

for some Lebesgue measure zero set $\mathcal{N}$,

and

Proof. For $\tilde{z}\in E(r_1, \lambda)$ and $\rho \in \left[\frac{\rho_0}{10}, \rho_0 \right)$, by (3.2) and (3.3), we derive

To attain the last bound, we consider two cases $p < 2$ and $p \geqslant 2$. When $p \geqslant2$, we see $\frac{p}{d} = 2$ and so

When $p < 2$, we see $\frac{p}{d} = \frac{(p-2)(n+2)}{2}+2$ and $\rho \geqslant \frac{\lambda^{\frac{p-2}2} (r_2-r_1)}{10}$ and so

Moreover, from the parabolic Lebesgue differentiation theorem, we deduce that, for almost every $\tilde{z}\in E(r_1, \lambda)$,

Since the map $\rho \mapsto \mathit{{\rlap{-} \smallint }}_{Q^\lambda_{\rho}(\tilde{z})} \left[|Du|^p + \left|\frac{F}{\delta}\right|^p \right]\, dz$ is continuous, there exists $\rho_{\tilde{z}}\in \left(0, \frac{r_2-r_1}{10}\right)$ such that

and

Hence we apply Vitali's covering lemma for $\{Q^\lambda_{\rho_{\tilde{z}}}(\tilde{z}): \tilde{z} \in E(r_1, \lambda) \}$ to complete the proof.

From now on, let us set for $i = 1, 2, 3, \dots$,

Step 2. (Estimates of super-level sets)

With the result in Lemma 3.1, we first estimates the Lebesgue measure of super-level set

where $A \geqslant 1$ will be determined below in (3.9), by using estimates in Lemma 2.1. Note from (3.5) that

We consider the following rescaled functions:

with $z = (x, t) \in Q_{2}$. Then it is obvious that $\mathbf{a}_{\lambda}(\xi)$ satisfies (1.2) and (1.3) with $\Omega_T = Q_2(0) = Q_2$. Then we see that $u_{\lambda, i}$ is a weak solution to

Moreover we have from (3.6) that

which implies

In addition, let $\widetilde{h}_{\lambda, i}$ be a weak solution to

Now, we consider sufficiently small constant $\varepsilon > 0$ which will be determined below in (3.17). Then by applying Lemma 2.1, one can find $\delta = \delta(n, \nu, L, p, \varepsilon) > 0$ satisfying (3.7) such that

Remark that both $\delta$ and $c_{\mathrm{Lip}}$ are independent of $\lambda$ and $i$. Therefore setting

where $z_i = (y_i, \tau_i)$, we obtain

We set

Then since

we have from the estimates in (3.8) that

which implies

Step 3. (weighted estimates of supper-level sets)

In this step, we estimate the weighted measure of super-level set

We first observe from (3.1) and (3.4) that

hence

This and the fact $\lambda \geqslant 1$ from (3.3) imply $Q^{(1)}_{i}\in \mathcal C_p$ for every $i = 1, 2, 3, \dots$. Then we obtain from (2.3) and (3.10) that

By Proposition 2.1, $w$ is a $p$-intrinsic $A_{q'}$ for some $q'\in (1, q)$. Now we suppose that

Then by (3.4) and (2.2) with $q$ replaced by $q'$ we have

where $c_0: = (2^{q'+1}[w]_{q'})^{\frac{1}{pq'}}$. Note that the right hand side is finite by the assumptions $F\in L^{pq}_{w, \mathrm{loc}}(\Omega_T)$ and (3.12). The above estimate means

Therefore, using Lemma 3.1, (3.11), (2.3) and (3.13), we obtain

Step 4. (A priori estimates)

We prove the estimate (1.7) under the additional assumption

where $2r < R_0$ and $R_0$ satisfies (3.1). Note that (3.15) implies (3.12).

Fix any $r \leqslant r_1 < r_2 \leqslant 2r$. Observe that

where $A$ and $B$ are from (3.9) and (3.3). We estimate the second term $I$. Applying (3.14), we derive

In the last inequality we apply the following elementary identity with $g = c_0|Du|$ or $\frac{c_0|F|}{\delta}$, $\beta_2 = pq$, $\beta_1 = pq'$ and $U = Q_{r_2}$:

Inserting the estimate for $I$ into (3.16) and recalling the definitions of $A$ and $B$ and the fact that $\varepsilon\in(0, 1)$, we have

where the constants $c_*$, $\gamma_1$ and $c$ depend on $n, p, \nu, L, q$ and $[w]_q$. At this stage, we choose $\varepsilon = \varepsilon(n, p, \nu, L, q, [w]_q)$ such that

hence $\delta$ is also determined as a small constant depending on $n, p, \nu, L, q$ and $[w]_q$. Therefore we obtain

for every $r \leqslant r_1 < r_2 \leqslant 2r$. Finally, applying Lemma 3.2 below with $\Psi(\rho) = \int_{Q_{\rho}}|Du|^{pq}w\, dz$ with $R_1 = r$ and $R_2 = 2r$ and recalling (3.2), we have that

This together with (2.3) implies (1.7).

Lemma 3.2 (Lemma 6.1 in ^[22]). Let $\Psi : [R_1, R_2] \rightarrow [0, \infty)$ be a bounded function. Suppose that for any $r_1$ and $r_2$ with $0 < R_1 \leqslant r_1 < r_2 \leqslant R_2$,

where $C > 0$ and $D \geqslant 0$, $\kappa > 0$ and $\vartheta \in [0, 1)$. Then there exists $c = (\vartheta, \kappa) > 0$ such that

Step 5. (Approximation)

Finally, we remove the a priori assumption (3.15) by a standard approximation argument. Suppose $w$ is a $p$-intrinsic $A_q$ weight and $F\in L^{pq}_{w, \mathrm{loc}}(\Omega_T, \mathbb{R}^n)$. Fix any $Q_{2r} = Q_{2r}(z_0)\Subset \Omega_T$ with $2r < R_0$. Then there exists $Q_R = Q_R(z_0)$ such that $Q_{2r}\Subset Q_R \Subset \Omega_T$. Note that by Proposition 2.1 (4), $w$ is a $p$-intrinsic $A_{pq}$ weight hence a usual parabolic $A_{pq}$ weight. Therefore, $C^\infty_{\mathrm c}(\mathbb{R}^{n+1})$ is dense in $L^{pq}_w(\mathbb{R}^{n+1})$, see e.g., [30, Lemma 2.1]. Therefore there exist $F_k\in C^\infty_{\mathrm c}(\mathbb{R}^{n+1}, \mathbb{R}^n)$, $k = 1, 2, 3, \dots$, such that

hence by (2.2),

We further assume that

Let $u_k \in C^0(t_0-R^2, t_0+R^2; L^2(B_R(x_0)) \cap L^p(t_0-R^2, t_0+R^2; W^{1, p}(B_R(x_0))$ be the unique weak solution to

see e.g., [34, Section III.4] for the existence of such $u_k$. In view of ^[2], we have at least $|Du_k|^p\in L^{\gamma}_{\mathrm{loc}}(Q_R)$ for every $\gamma > 1$ since $|F_k|^p\in L^{\gamma}(Q_{R})$. In particular, by Hölder's inequality with Proposition 2.1 (2),

which implies the a priori assumption in (3.15) for $u_k$ hence it follows from the previous results in Step 4 that

Now we take $u-u_k$ as a test function in the weak forms of (1.1) and (3.20) to get

Then, in a similar way as in the proof of Lemma 2.1, we derive

for any $\tau_1 \in (0, 1)$, by applying Young's inequality.

If $p \geqslant 2$, since $|Du_k-Du|^p \leqslant (|Du_k|^2 + |Du|^2)^{\frac{p-2}{2}} |Du_k-Du|^2$, we infer

by taking sufficiently small $\tau_1 > 0$ and (3.19). If $\frac{2n}{n+2} < p < 2$, applying Young's inequality, we have that for any $\tau_2\in(0, 1)$,

and then by taking sufficiently small $\tau_1, \tau_2 > 0$ and (3.19),

Eventually, for any $p > \frac{2n}{n+2}$, we obtain that

Moreover, from (3.18) we see

Then taking into account (3.22) with (3.23),

Since $\tau_1\in(0, 1)$ is arbitrary, we have that

Now, if $\frac{2n}{n+2} < p < 2$, the Hölder inequality yields

and therefore by virtue of (3.23), we obtain

This also holds in case $p \geqslant 2$ from (3.22), because

In turn, we obtain that for every $p > \frac{2n}{n+2}$

In particular, we also have that $Du_k \longrightarrow Du$ a.e. in $L^{p}(Q_{2r}, \mathbb{R}^n)$ as $k\to \infty$, up to subsequence.

Finally by passing $k\to\infty$ from (3.21) and applying the above convergence results for $F_k$ and $Du_k$ with Fatou's lemma, we obtain

Therefore, we complete the proof.

Acknowledgements

We would like to thank the referees for many helpful comments. M. Lee was supported by the National Research Foundation of Korea by the Korean Government (NRF-2021R1A4A1032418). J. Ok was supported by the National Research Foundation of Korea by the Korean Government (NRF-2022R1C1C1004523).

Conflict of interest

The authors declare no conflict of interest.