Generalized Local Morrey Spaces Associated with Ball Banach Function Spaces and Their Application

Abstract

This paper is devoted to the analysis of boundedness for fractional integral operators, Calderón–Zygmund singular integral operators, and their corresponding commutators on generalized local Morrey spaces associated with ball Banach function spaces. These foundational results are then applied to establish the local regularity within the $L{M}_X^{\phi }$ Morrey spaces for the solution gradients of second-order elliptic equations expressed in divergence form.

1. Introduction

Our work is concerned with the study of second-order elliptic equations in divergence form,

$$\mathcal{L}u:=\sum _{i,j=1}^n{\left(a_{ij}\left(x\right)u_{x_i}\right)}_{x_j}=\nabla ·f\mathrm{for}\mathrm{almost}\mathrm{all}x$$

(1)

posed in a bounded region $\mathrm{\Omega }\subset {\mathbb{R}}^n$ of dimension $n\ge 3$.

A central class of elliptic equations in divergence form has been the subject of deep and longstanding investigation. A canonical problem that has driven much of the theory is given by

$$\left\{\begin{array}{cc}Lu=-\mathrm{div}A\nabla u=f\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \mathrm{on}\partial \mathrm{\Omega },\hfill \end{array}\right.$$

Here, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded open set, and the $n\times n$ matrix $A\left(x\right)=\left(a_{ij}\left(x\right)\right)$ contains real-valued, measurable coefficient functions. The essential ellipticity requirement posits the existence of constants $\lambda ,\mathrm{\Lambda }>0$ for which, for almost all $x\in \mathrm{\Omega }$ and any $\xi \in {\mathbb{R}}^n$,

$${\lambda \left|\xi \right|}^2\le \sum _{i,j=1}^n{a}_{ij}\left(x\right){\xi }_i{\xi }_j\le \mathrm{\Lambda }{\left|\xi \right|}^2.$$

The classical $L^p$ theory developed by Gilbarg and Trudinger [1] remains applicable when A is continuous and $\partial \mathrm{\Omega }\in C^{2,\alpha }$. Subsequent advancements in function space and operator theory have led to notable progress in the regularity theory for solutions of (1). As documented in [2,3], the known regularity results in classical Lebesgue spaces have been successfully generalized to a broader class of function spaces, such as Morrey spaces, Herz spaces, and variable-exponent Herz spaces; see, for example, [4,5,6].

Recent years have witnessed a growing interest in the study of ball Banach function spaces. The classical theory of Banach function spaces, originally introduced in [7], encompasses well-known spaces like Lebesgue spaces, Lorentz spaces, and variable-exponent Lebesgue spaces. Building upon this foundation, Sawano, Ho, Yang, and Yang [4] subsequently generalized this framework to define ball Banach function spaces and introduced the associated Hardy spaces.

The exploration of ball quasi-Banach function spaces has proliferated, yielding significant work on associated Sobolev spaces [8,9], Hardy spaces [4,10], Morrey spaces [11], and BMO spaces [12,13]. The boundedness of integral operators within this context has been independently and closely pursued by many researchers [14,15], not least due to its critical applications. These applications primarily involve employing the boundedness properties of singular integral operators to advance the regularity theory for partial differential equations, potentially featuring discontinuous coefficients, which remains an area of active and considerable interest.

In the theory of partial differential equations, establishing the existence, uniqueness, and regularity of solutions constitutes a fundamental aspect of analysis, necessitating their study within various function spaces. The inclusion relations inherent to ball Banach function spaces offer a structural framework for elucidating solution behavior. This methodological framework allows one to commence from a specific function space known to contain the solution and then progressively infer its properties in other spaces via these inclusions, thereby achieving a comprehensive and unified understanding of the solution’s attributes across different functional settings.

The present work is devoted to generalizing the regularity results for solutions of (1) to generalized local Morrey spaces within the framework of ball Banach function spaces. We consider a linear elliptic operator $\mathcal{L}$ whose coefficients $a_{ij}$ are in $VMO$. The problem is postulated under the condition that the vector field $f=(f_1,f_2,\dots ,f_n)$ has each component $f_i$ in the space $L{M}_X^{\phi }$ for $i=1,\dots ,n$ (definitions are provided in Section 3).

The structure of this paper is organized as follows: Section 3 introduces the necessary preliminaries, including definitions of ball Banach function spaces X and the associated generalized local Morrey spaces $L{M}_X^{\phi }$, along with two key assumptions on the differential operator’s coefficients. Included in this section is a discussion of the John–Nirenberg space of functions with bounded mean oscillation and the Sarason class of functions with vanishing mean oscillation. The boundedness of linear operators stemming from fractional integrals, Calderón–Zygmund singular integrals, and their commutators in $L{M}_X^{\phi }$ is established in Section 4 and Section 3. Finally, Section 5 is dedicated to proving the regularity of the first-order derivatives of solutions to divergence-form elliptic equations within the $L{M}_X^{\phi }$ framework.

The following notations are employed in this paper:

• We denote by c and C positive constants, which are independent of the main parameters and may vary from one occurrence to another.
• We write $B_1\sim B_2$ if there exists a constant $C>1$ such that $C^{-1}\le B_1/B_2\le C$. The inequality $B_1\lesssim B_2$ is used to denote the relation $B_1\le C{B}_2$.
• For $r>0$ and $x_0\in {\mathbb{R}}^n$, let $B(x_0,r)=\{y:|y-x_0|<r\}$ denote the open ball of radius r centered at x, and let $\mathbb{B}=\{B(x,r):x\in {\mathbb{R}}^n,r>0\}$ be the family of all such balls.
• For a measurable set $E\subseteq {\mathbb{R}}^n$, the characteristic function, Lebesgue measure, and complementary set are denoted by ${\chi }_E$, $\left|E\right|$, and $E^c$, respectively.
• We denote by $\mathcal{M}\left({\mathbb{R}}^n\right)$ the class of all Lebesgue measurable functions defined on ${\mathbb{R}}^n$.

2. Some Preliminaries and Notations

We begin by recalling the definitions of ball Banach function spaces and their associated spaces.

Definition 1

(cf. [4]). A Banach space $X\subseteq \mathcal{M}\left({\mathbb{R}}^n\right)$ is classified as a ball Banach function space if it satisfies the following axioms:

(i) 

Positivity: ${\parallel f\parallel }_X=0$ if and only if $f=0$ almost everywhere.

(ii) 

Lattice property: If $\left|g\right|\le \left|f\right|$ almost everywhere, then ${\parallel g\parallel }_X\le {\parallel f\parallel }_X$.

(iii) 

Fatou property: If $0\le f_m\uparrow f$ almost everywhere, then ${\parallel f_m\parallel }_X\uparrow {\parallel f\parallel }_X$.

(iv) 

Local integrability: For every ball $B\in \mathbb{B}$, the characteristic function ${\chi }_B$ belongs to X.

(v) 

Norm-localization: For every $B\in \mathbb{B}$, there exists a constant $C\left(B\right)>0$ such that

$${\int }_B\left|f\left(x\right)\right|dx\le {C\left(B\right)\parallel f\parallel }_X\mathit{for}\mathit{all}f\in X.$$

Lemma 1

(See [4]). Suppose X is a ball Banach function space and $X^{\prime }$ is its associate space. Then, for all $f\in X$ and $g\in X^{\prime }$, the product $fg$ belongs to $L^1\left({\mathbb{R}}^n\right)$, and it holds that

$${\int }_{{\mathbb{R}}^n}\left|f\left(x\right)g\left(x\right)\right|dx\le {\parallel f\parallel }_X{\parallel g\parallel }_{X^{\prime }}.$$

(2)

Definition 2

(See [16]). The space $\mathrm{BMO}\left({\mathbb{R}}^n\right)$ of functions with bounded mean oscillation is defined as the collection of all locally integrable functions f on ${\mathbb{R}}^n$ for which the BMO-norm

$${\parallel f\parallel }_{\mathrm{BMO}}=\underset{B\in \mathbb{B}}{sup}{\displaystyle \frac{1}{\left|B\right|}}{\int }_B|f\left(x\right)-f_B|dx$$

is finite. Here, $f_B={\displaystyle \frac{1}{\left|B\right|}}{\int }_{B}f\left(y\right)dy$ is the mean value of f on the ball B.

Let $f\in \mathrm{BMO}\left({\mathbb{R}}^n\right)$. We define its VMO modulus ${\eta }_f\left(r\right)$ for $r>0$ by

$${\eta }_f\left(r\right)=sup\{{\displaystyle \frac{1}{\left|B\right|}}{\int }_B|f-f_B|:B\mathrm{is}\mathrm{a}\mathrm{ball}\mathrm{of}\mathrm{radius}\rho \le r\}.$$

The function f is said to belong to the Sarason space $\mathrm{VMO}\left({\mathbb{R}}^n\right)$ (see [17]) if and only if its VMO modulus vanishes at zero, i.e., ${lim}_{r\to 0}{\eta }_f\left(r\right)=0$.

Definition 3

(See [18]). For a function $f\in \mathcal{M}\left({\mathbb{R}}^n\right)$, its ${\mathrm{BMO}}_X$ norm is defined by

$${\parallel f\parallel }_{{\mathrm{BMO}}_X}:=\underset{B\in \mathbb{B}}{sup}{\displaystyle \frac{{∥{\chi }_B(f-f_B)∥}_X}{{∥{\chi }_B∥}_X}},$$

where the supremum is taken over all balls $B=B(x_0,r)$ in ${\mathbb{R}}^n$.

A result from [18,19] ensures the coincidence ${\mathrm{BMO}}_X=\mathrm{BMO}$ whenever the Hardy–Littlewood maximal operator exhibits boundedness on the associate space of X.

The primary objective of this work is to analyze the regularity properties of solutions to elliptic equations within the framework of generalized local Morrey spaces associated with ball Banach function spaces. Toward this end, we first recall the following necessary definitions:

Definition 4.

Given a ball Banach function space X and a measurable function $\phi (x,r)>0$ that is non-increasing in r for each fixed $x\in {\mathbb{R}}^n$, the generalized local Morrey space $L{M}_X^{\phi }\left({\mathbb{R}}^n\right)$ consists of all functions $f\in \mathcal{M}\left({\mathbb{R}}^n\right)$ satisfying

$${\parallel f\parallel }_{L{M}_X^{\phi }}:=\underset{B(x_0,r)}{sup}{\displaystyle \frac{\parallel {\chi }_{B(x_0,r)}{f\parallel }_X}{\phi (x_0,r){\parallel {\chi }_{B(x_0,r)}\parallel }_X}}<\infty ,$$

where the supremum is taken over all balls $B(x_0,r)\subset {\mathbb{R}}^n$.

Definition 5.

Given a ball Banach function space X and a measurable function $\phi (x,r)>0$, the modified generalized local Morrey space ${\widetilde{LM}}_X^{\phi }\left({\mathbb{R}}^n\right)$ is introduced as the collection of functions $f\in \mathcal{M}\left({\mathbb{R}}^n\right)$ such that

$${\parallel f\parallel }_{{\widetilde{LM}}_X^{\phi }}:={\parallel f\parallel }_{L{M}_X^{\phi }}+{\parallel f\parallel }_X<\infty .$$

In this section, we will utilize the following result concerning the boundedness properties of the weighted Hardy operator:

$$H_w^{\ast }g\left(t\right):={\int }_t^{d}g\left(s\right)w\left(s\right)ds,\mathrm{for}0<t<d<\infty ,$$

where the weight function w is a non-negative measurable function defined on the interval $(0,d)$.

The boundedness of the weighted Hardy operators $H_w$ and $H_w^{\ast }$, whose proofs are detailed in [20,21], is summarized as follows:

Lemma 2.

The following criterion holds: the inequality

$$\parallel v_2{H}_w{g\parallel }_{L^{\infty }}\le C{\parallel v_{1}g\parallel }_{L^{\infty }}$$

is satisfied for every non-negative and non-increasing function g if and only if

$$A=\underset{t>0}{sup}{v}_2\left(t\right){\int }_t^{\infty }{\displaystyle \frac{w\left(s\right)ds}{\parallel v_1{\parallel }_{L^{\infty }(s,\infty )}}}<\infty .$$

Moreover, the smallest possible constant C in this estimate is such that $C\sim A$.

Lemma 3.

We have the following characterization for the boundedness of $H_w^{\ast }$: the inequality

$$\underset{t>0}{ess\; sup}{v}_2\left(t\right)H_w^{\ast }g\left(t\right)\le C\underset{t>0}{ess\; sup}{v}_1\left(t\right)g\left(t\right)$$

is valid for all non-negative, non-increasing functions g if and only if

$$A=\underset{t>0}{sup}{v}_2\left(t\right){\int }_t^{\infty }\left(1+ln{\displaystyle \frac{s}{t}}\right){\displaystyle \frac{w\left(s\right)}{{ess\; sup}_{s<\tau <\infty }{v}_1\left(\tau \right)}}ds<\infty .$$

Moreover, the best constant C in this inequality is comparable to A.

3. Fractional Integral Operators

The fractional integral operator $I_{\alpha }$ of order $0<\alpha <n$ acts on a locally integrable function f via the integral

$$I_{\alpha }f\left(x\right)={\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(z\right)}{{|x-z|}^{n-\alpha }}}dz,x\in {\mathbb{R}}^n.$$

This operator plays a fundamental role in harmonic analysis.

Lemma 4

(See [22]). Let $0<\alpha <n$, and let ${\phi }_1:{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$. Suppose X and Y are ball Banach function spaces satisfying, for every ball $B\in \mathbb{B}$,

$${\parallel {\chi }_B\parallel X^{\prime }\parallel {\chi }_B\parallel }_Y\sim {\left|B\right|}^{1-\alpha /n}.$$

(3)

Under the assumption that $I_{\alpha }$ is bounded from X to Y, the estimate

$$\parallel {\chi }_{B(z_0,r)}{I}_{\alpha }{f\parallel }_Y\lesssim {\parallel {\chi }_{B(z_0,r)}\parallel }_Y{\int }_{2r}^{\infty }{\displaystyle \frac{\parallel {\chi }_{B(z_0,t)}{f\parallel }_X}{\parallel {\chi }_{B(z_0,t)}{\parallel }_Y}}{\displaystyle \frac{dt}{t}}$$

holds for any $f\in M_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$.

Theorem 1.

Let $0<\alpha <n$, and let $X,Y$ be ball Banach function spaces satisfying (3) for all $B\in \mathbb{B}$. Suppose the fractional integral operator $I_{\alpha }$ is bounded from X to Y, and let ${\phi }_1,{\phi }_2:{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$ be functions such that the balance condition

$${\int }_r^{\infty }{\displaystyle \frac{\mathrm{ess}{\inf }_{t<s<\infty }{\phi }_1(x_0,s){\parallel {\chi }_{B(x_0,s)}\parallel }_X}{\parallel {\chi }_{B(x_0,t)}{\parallel }_Y}}{\displaystyle \frac{dt}{t}}\lesssim {\phi }_2(x_0,r)$$

(4)

holds. Then for any $f\in L{M}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$, we have the boundedness

$$\parallel I_{\alpha }{f\parallel }_{L{M}_Y^{{\phi }_2}}\lesssim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}}.$$

Proof. 

From Lemma 2 and Lemma 4 with the assignments $v_2\left(t\right)={\phi }_2{(z_0,t)}^{-1}$, $v_1\left(t\right)={\phi }_1{(z_0,t)}^{-1}{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}$, $g\left(t\right)=\parallel {\chi }_{B(z_0,t)}{f\parallel }_X$, and $w\left(t\right)=t^{-1}{\parallel {\chi }_{B(z_0,t)}\parallel }_Y^{-1}$, it follows that

$$\begin{array}{cc}\hfill \parallel I_{\alpha }{f\parallel }_{L{M}_Y^{{\phi }_2}}& =\underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}\parallel {\chi }_{B(z_0,r)}{\parallel }_Y^{-1}{\parallel {\chi }_{B(z_0,r)}{I}_{\alpha }f\parallel }_Y\hfill \\ & \lesssim \underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}{\int }_r^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_Y^{-1}{\displaystyle \frac{dt}{t}}\hfill \\ & \lesssim \underset{r>0}{sup}{\phi }_1{(z_0,r)}^{-1}\parallel {\chi }_{B(z_0,r)}{\parallel }_X^{-1}\parallel {\chi }_{B(z_0,r)}{f\parallel }_X\sim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}}.\hfill \end{array}$$

The proof is finished.  □

Example 1.

Interestingly, when $X=L^p\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), Theorem 1 reduces to the result of [3]; when $X=L^{p(·)}\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), it reduces to the result of [23]; and when $X=L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), it reduces to the result of [24].

Theorem 2.

Let $0<\alpha <n$, and let $X,Y$ be ball Banach function spaces satisfying condition (3) for all balls $B\in \mathbb{B}$. Assume that the fractional integral operator $I_{\alpha }$ is bounded from X to Y, and that the functions ${\phi }_1,{\phi }_2:{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$ satisfy condition (4). Then for any function f in the modified generalized local Morrey space ${\widetilde{LM}}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$, we establish the boundedness

$$\parallel I_{\alpha }{f\parallel }_{{\widetilde{LM}}_Y^{{\phi }_2}}\lesssim {\parallel f\parallel }_{{\widetilde{LM}}_X^{{\phi }_1}}.$$

Proof. 

The proof can be given via adapting the argument of Theorem 1 and utilizing the definition of ${\widetilde{LM}}_X^{\phi }$.  □

Example 2.

Interestingly, when $X=L^p\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), Theorem 2 reduces to the result of [3].

4. Calderón–Zygmund Operators and Commutators

We call a linear operator T a Calderón–Zygmund singular integral operator if it is bounded on $L^2\left({\mathbb{R}}^n\right)$ and satisfies $T\left(C_c^{\infty }\left({\mathbb{R}}^n\right)\right)\subset L_{\mathrm{loc}}^1\left({\mathbb{R}}^n\right)$. For further details, we refer the reader to [25]. Its action on any test function $f\in C_c^{\infty }\left({\mathbb{R}}^n\right)$ is given by the integral representation

$$Tf\left(x\right)={\int }_{{\mathbb{R}}^n}k(x,y)f\left(y\right)dy,$$

(5)

which is valid for all x not in the support of f.

The kernel $k(x,y)$ is a function defined for $x\ne y$. It is said to satisfy the standard Calderón–Zygmund estimates if there exist constants $C>0$ and $0<ϵ\le 1$ for which

• Size condition: For all $x\ne y$,

  $$\left|k(x,y)\right|\le {\displaystyle \frac{C}{{|x-y|}^n}}.$$

  (6)
• Smoothness condition: Whenever $2|x-x^{\prime }|\le |x-y|$,

  $$|k(x,y)-k(x^{\prime },y)|+|k(y,x)-k(y,x^{\prime })|\le {\displaystyle \frac{C|x-x^{\prime }{|}^{ϵ}}{{|x-y|}^{n+ϵ}}}.$$

Theorem 3.

Consider the following assumptions:

• T is a linear operator defined by (5) and satisfying the kernel estimate (6);
• X is a ball Banach function space, and both the Hardy–Littlewood maximal operator M and T are bounded on X;
• ${\phi }_1,{\phi }_2:{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$ satisfy the balance condition

  $${\int }_r^{\infty }{\displaystyle \frac{{ess\; inf}_{t<s<\infty }{\phi }_1(x_0,s){\parallel {\chi }_{B(x_0,s)}\parallel }_X}{\parallel {\chi }_{B(x_0,t)}{\parallel }_X}}{\displaystyle \frac{dt}{t}}\lesssim {\phi }_2(x_0,r);$$

  (7)

• it follows that for any $f\in L{M}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$,

  $${\parallel Tf\parallel }_{L{M}_X^{{\phi }_2}}\lesssim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}}.$$

Proof. 

In view of (2) and $\parallel {\chi }_{B(z_0,t)}{\parallel }_{X^{\prime }}\lesssim t^n{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}$, for any $f\in L{M}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$ and $x\in B(z_0,r)$, we have

$$\begin{array}{cc}\hfill \left|Tf\right(x\left)\right|& \le |T\left({\chi }_{B(z_0,2r)}f\right)\left(x\right)|+{\int }_{B^c(z_0,2r)}\left|k(x,y)f\left(y\right)\right|dy\hfill \\ & \lesssim |T\left({\chi }_{B(z_0,2r)}f\right)\left(x\right)|+{\int }_{2r}^{\infty }{\displaystyle \frac{\left|f\right(y\left)\right|}{|z_0-y|}}dy\hfill \\ & \lesssim |T\left({\chi }_{B(z_0,2r)}f\right)\left(x\right)|+{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{\prime }{\displaystyle \frac{dt}{t^{n+1}}}\hfill \\ & \lesssim |T\left({\chi }_{B(z_0,2r)}f\right)\left(x\right)|+{\int }_{2r}^{\infty }{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\hfill \\ & \lesssim |T\left({\chi }_{B(z_0,2r)}f\right)\left(x\right)|+{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,2r)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.\hfill \end{array}$$

Therefore,

$$\parallel {\chi }_{B(z_0,r)}{Tf\parallel }_X\lesssim \parallel {\chi }_{B(z_0,r)}T\left({\chi }_{B(z_0,2r)}f\right){\parallel }_X+\parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.$$

(8)

From ${\chi }_{B(z_0,2r)}f\in X$ and the boundedness of T on X, we have

$$\parallel {\chi }_{B(z_0,r)}T\left({\chi }_{B(z_0,2r)}f\right){\parallel }_X\le \parallel T\left({\chi }_{B(z_0,2r)}f\right){\parallel }_X\lesssim {\parallel {\chi }_{B(z_0,2r)}f\parallel }_X.$$

By using (2) and $\parallel {\chi }_{B(z_0,t)}{\parallel }_{X^{\prime }}\lesssim t^n{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}$, there holds

$$\begin{array}{cc}\hfill \parallel {\chi }_{B(z_0,2r)}{f\parallel }_X& \sim |B(z_0,r)|\parallel {\chi }_{B(z_0,2r)}{f\parallel }_X{\int }_{2r}^{\infty }{\displaystyle \frac{dt}{t^{1+n}}}\hfill \\ & \lesssim |B(z_0,r)|{\int }_{2r}^{\infty }{\parallel {\chi }_{B(z_0,t)}f\parallel }_X{\displaystyle \frac{dt}{t^{1+n}}}\hfill \\ & \lesssim \parallel {\chi }_{B(z_0,r)}{\parallel }_X\parallel {\chi }_{B(z_0,r)}{\parallel }_X^{\prime }{\int }_{2r}^{\infty }{\parallel {\chi }_{B(z_0,t)}f\parallel }_X{\displaystyle \frac{dt}{t^{1+n}}}\hfill \\ & \lesssim \parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{\prime }{\displaystyle \frac{dt}{t^{1+n}}}\hfill \\ & \lesssim \parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.\hfill \end{array}$$

As a consequence,

$$\parallel {\chi }_{B(z_0,r)}T\left({\chi }_{B(z_0,2r)}f\right){\parallel }_X\lesssim \parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.$$

(9)

Combining (8) and (9), we arrive at

$$\parallel {\chi }_{B(z_0,r)}{Tf\parallel }_X\lesssim \parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.$$

By using Lemma 2 with $v_2\left(t\right)={\phi }_2{(z_0,t)}^{-1}$, $v_1\left(t\right)={\phi }_1{(z_0,t)}^{-1}{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}$, $g\left(t\right)=\parallel {\chi }_{B(z_0,t)}{f\parallel }_X$ and $w\left(t\right)=t^{-1}{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}$, we have

$$\begin{array}{cc}\hfill \underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}& {\int }_r^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\hfill \\ & \lesssim \underset{r>0}{sup}{\phi }_1{(z_0,r)}^{-1}\parallel {\chi }_{B(z_0,r)}{\parallel }_X^{-1}\parallel {\chi }_{B(z_0,r)}{f\parallel }_X\sim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}}<\infty .\hfill \end{array}$$

Then, it yields

$$\begin{array}{cc}\hfill {\parallel Tf\parallel }_{L{M}_X^{{\phi }_2}}& =\underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}\parallel {\chi }_{B(z_0,r)}{\parallel }_X^{-1}{\parallel {\chi }_{B(z_0,r)}Tf\parallel }_X\hfill \\ & \lesssim \underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}{\int }_r^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X\parallel {\chi }_{B(z_0,t)}{\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\lesssim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}}.\hfill \end{array}$$

The establishment of the theorem is now complete.  □

Example 3.

Interestingly, when $X=L^p\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), Theorem 3 reduces to the result of [3]; when $X=L^{p(·)}\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), it reduces to the result of [26].

Theorem 4.

Assume the following:

• T is a linear operator given by (5) and satisfying (6);
• X is a ball Banach function space with both M and T bounded on X;
• ${\phi }_1,{\phi }_2:{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$ satisfy condition (7).

• Then, for any $f\in {\widetilde{LM}}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$, the operator T satisfies the boundedness property:

  $${\parallel Tf\parallel }_{{\widetilde{LM}}_X^{{\phi }_2}}\lesssim {\parallel f\parallel }_{{\widetilde{LM}}_X^{{\phi }_1}}.$$

Proof. 

The proof can be given via adapting the argument of Theorem 3 and utilizing the definition of ${\widetilde{LM}}_X^{\phi }$.  □

The commutator between a locally integrable function b and a linear operator T is the operator $T_b$ given by means of the singular integral

$$T_{b}f\left(x\right)={\int }_{{\mathbb{R}}^n}(b\left(x\right)-b\left(y\right))k(x,y)f\left(y\right)dy.$$

(10)

Example 4.

Interestingly, when $X=L^p\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), Theorem 4 reduces to the result of [3].

The systematic study of commutators forms an important part of harmonic analysis. Among these, the commutator $T_b$ generated by Calderón–Zygmund operators is defined similarly to T itself.

Lemma 5.

Suppose $f\in {\mathrm{BMO}}_X\left({\mathbb{R}}^n\right)$. Then, for any $x\in {\mathbb{R}}^n$ and for any radii $r,t>0$ with $2r<t$, the difference of the mean values satisfies

$$|f_{B(x,r)}-f_{B(x,t)}{|\lesssim \parallel f\parallel }_{{\mathrm{BMO}}_X}ln{\displaystyle \frac{t}{r}}.$$

(11)

Theorem 5.

Assume the following conditions hold:

• $T_b$ is the linear commutator defined by (10) and satisfies the kernel estimate (6);
• X is a ball Banach function space, and the Hardy–Littlewood maximal operator M is bounded on X and $X^{\prime }$;
• $T_b$ is bounded on X;
• $b\in {\mathrm{BMO}}_X\left({\mathbb{R}}^n\right)$;
• ${\phi }_1,{\phi }_2:{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$ satisfy the balance condition

  $${\int }_r^{\infty }\left(1+ln{\displaystyle \frac{t}{r}}\right){\displaystyle \frac{{ess\; inf}_{t<s<\infty }{\phi }_1(x_0,s){\parallel {\chi }_{B(x_0,s)}\parallel }_X}{\parallel {\chi }_{B(x_0,t)}{\parallel }_X}}{\displaystyle \frac{dt}{t}}\lesssim {\phi }_2(x_0,r).$$

  (12)

• Then, the commutator $T_b$ is bounded from $L{M}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$ to $L{M}_X^{{\phi }_2}\left({\mathbb{R}}^n\right)$, and moreover,

  $$\parallel T_b{f\parallel }_{L{M}_X^{{\phi }_2}}\lesssim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}},\forall f\in L{M}_X^{{\phi }_1}\left({\mathbb{R}}^n\right).$$

Proof. 

For any $f\in L{M}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$ and $x\in B(z_0,r)$, we have

$$\left|T_{b}f\left(x\right)\right|\le \left|T_b\left({\chi }_{B(z_0,2r)}f\right)\left(x\right)\right|+{\int }_{B^c(z_0,2r)}\left|(b\left(x\right)-b\left(y\right))k(x,y)f\left(y\right)\right|dy$$

Since for $x\in B(z_0,r)$ and $y\in B^c(z_0,2r)$, ${\displaystyle \frac{1}{2}}|z_0-y|\le |x-y|\le {\displaystyle \frac{3}{2}}|z_0-y|$, conditions (6) and (10) assure

$$\begin{array}{cc}& {\int }_{B^c(z_0,2r)}\left|(b\left(x\right)-b\left(y\right))k(x,y)f\left(y\right)\right|dy\le {\int }_{B^c(z_0,2r)}{\displaystyle \frac{\left|b\right(x)-b(y\left)\right|}{{|x-y|}^n}}\left|f\left(y\right)\right|dy\hfill \\ & \le {\int }_{B^c(z_0,2r)}{\displaystyle \frac{\left|b\right(x)-b(y\left)\right|}{{|x-y|}^n}}\left|f\left(y\right)\right|dy\lesssim {\int }_{B^c(z_0,2r)}{\displaystyle \frac{\left|b\right(x)-b(y\left)\right|}{|z_0{-y|}^n}}\left|f\left(y\right)\right|dy\hfill \\ & \lesssim |b\left(x\right)-b_{B(z_0,r)}|{\int }_{B^c(z_0,2r)}{\displaystyle \frac{\left|f\right(y\left)\right|}{|z_0{-y|}^n}}dy+{\int }_{B^c(z_0,2r)}{\displaystyle \frac{|b\left(y\right)-b_{B(z_0,r)}|}{|z_0{-y|}^n}}\left|f\left(y\right)\right|dy:=\mathrm{I}+\mathrm{II}.\hfill \end{array}$$

For term $\mathrm{II}$, we have

$$\begin{array}{cc}\hfill \mathrm{II}& ={\int }_{B^c(z_0,2r)}{\displaystyle \frac{|b_{B(z_0,r)}-b\left(y\right)|}{|z_0{-y|}^n}}\left|f\left(y\right)\right|dy\hfill \\ & \sim {\int }_{B^c(z_0,2r)}|b\left(y\right)-b_{B(z_0,r)}\left|\right|f\left(y\right)|{\int }_{|z-y|}^{\infty }{\displaystyle \frac{dt}{t^{n+1}}}dy\hfill \\ & \sim {\int }_{2r}^{\infty }{\int }_{2r\le |z_0-y|<t}|b\left(y\right)-b_{B(z_0,r)}\left|\right|f\left(y\right)|dy{\displaystyle \frac{dt}{t^{n+1}}}\hfill \\ & \lesssim {\int }_{2r}^{\infty }{\int }_{B(z_0,t)}|b\left(y\right)-b_{B(z_0,r)}\left|\right|f\left(y\right)|dy{\displaystyle \frac{dt}{t^{n+1}}}.\hfill \end{array}$$

By using (2) and (11), we further obtain

$$\begin{array}{cc}\hfill \mathrm{II}& \lesssim {\int }_{2r}^{\infty }{\int }_{B(z_0,t)}|b\left(y\right)-b_{B(z_0,t)}\left|\right|f\left(y\right)|dy{\displaystyle \frac{dt}{t^{n+1}}}\hfill \\ & +{\int }_{2r}^{\infty }{\int }_{B(z_0,t)}|b_{B(z_0,r)}-b_{B(z_0,t)}\left|\right|f\left(y\right)|dy{\displaystyle \frac{dt}{t^{n+1}}}\hfill \\ & \lesssim {\int }_{2r}^{\infty }\parallel (b-b_{B(z_0,t)}){\chi }_{B(z_0,t)}{\parallel }_X{\parallel {\chi }_{B(z_0,t)}f\parallel }_X{\displaystyle \frac{dt}{t^{n+1}}}\hfill \\ & +{\int }_{2r}^{\infty }|b_{B(z_0,r)}-b_{B(z_0,t)}|\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_{X^{\prime }}{\displaystyle \frac{dt}{t^{n+1}}}\hfill \\ & \lesssim {\int }_{2r}^{\infty }{\displaystyle \frac{\parallel (b-b_{B(z_0,t)}){\chi }_{B(z_0,t)}{\parallel }_{X^{\prime }}}{\parallel {\chi }_{B(z_0,t)}{\parallel }_X}}\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\hfill \\ & +{\int }_{2r}^{\infty }|b_{B(z_0,r)}-b_{B(z_0,t)}|\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\hfill \\ & \lesssim {\parallel b\parallel }_{{\mathrm{BMO}}_X}{\int }_{2r}^{\infty }\left(1+ln{\displaystyle \frac{t}{r}}\right)\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.\hfill \end{array}$$

As a consequence,

$$\begin{array}{cc}\hfill \parallel {\chi }_{B(z_0,r)}{T}_b{f\parallel }_X& \lesssim \parallel {\chi }_{B(z_0,r)}{T}_b\left({\chi }_{B(z_0,2r)}f\right){\parallel }_X\hfill \\ & +\parallel {\chi }_{B(z_0,r)}(b(·)-b_{B(z_0,r)}){\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\hfill \\ & +\parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\left(1+ln{\displaystyle \frac{t}{r}}\right)\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}=:{\mathrm{I}}_1+{\mathrm{I}}_2+{\mathrm{I}}_3.\hfill \end{array}$$

Since ${\chi }_{B(z_0,2r)}f\in X$ and $T_b$ is bounded on X, we get

$$\parallel {\chi }_{B(z_0,r)}{T}_b\left({\chi }_{B(z_0,2r)}f\right){\parallel }_X\le \parallel T_b\left({\chi }_{B(z_0,2r)}f\right){\parallel }_X\lesssim {\parallel {\chi }_{B(z_0,2r)}f\parallel }_X,$$

which together with (9) further imply

$${\mathrm{I}}_1\lesssim \parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.$$

From Definition 3, we obtain

$$\begin{array}{cc}\hfill {\mathrm{I}}_2& \lesssim {\displaystyle \frac{\parallel {\chi }_{B(z_0,r)}(b-b_{B(z_0,r)}){\parallel }_X}{\parallel {\chi }_{B(z_0,r)}{\parallel }_X}}\parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\hfill \\ & \lesssim {\parallel b\parallel }_{{\mathrm{BMO}}_X}\parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.\hfill \end{array}$$

As shown above, we arrive at

$$\parallel {\chi }_{B(z_0,r)}{T}_b{f\parallel }_X\lesssim \parallel {\chi }_{B(z_0,r)}{\parallel }_X{\int }_{2r}^{\infty }\left(1+ln{\displaystyle \frac{t}{r}}\right)\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}.$$

By using Lemma 3 with $v_2\left(t\right)={\phi }_2{(z_0,t)}^{-1}$, $v_1\left(t\right)={\phi }_1{(z_0,t)}^{-1}{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}$, $g\left(t\right)=\parallel {\chi }_{B(z_0,t)}{f\parallel }_X$ and $w\left(t\right)=t^{-1}{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}$, we have

$$\underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}{\int }_r^{\infty }\left(1+ln{\displaystyle \frac{t}{r}}\right)\parallel {\chi }_{B(z_0,t)}{f\parallel }_X{\parallel {\chi }_{B(z_0,t)}\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}$$

$$\lesssim \underset{r>0}{sup}{\phi }_1{(z_0,r)}^{-1}\parallel {\chi }_{B(z_0,r)}{\parallel }_X^{-1}\parallel {\chi }_{B(z_0,r)}{f\parallel }_X\sim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}}<\infty .$$

Then, it yields

$$\begin{array}{cc}\hfill \parallel T_b{f\parallel }_{L{M}_X^{{\phi }_2}}& =\underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}\parallel {\chi }_{B(z_0,r)}{\parallel }_X^{-1}{\parallel {\chi }_{B(z_0,r)}{T}_{b}f\parallel }_X\hfill \\ & \lesssim \underset{r>0}{sup}{\phi }_2{(z_0,r)}^{-1}{\int }_r^{\infty }\left(1+ln{\displaystyle \frac{t}{r}}\right)\parallel {\chi }_{B(z_0,t)}{f\parallel }_X\parallel {\chi }_{B(z_0,t)}{\parallel }_X^{-1}{\displaystyle \frac{dt}{t}}\lesssim {\parallel f\parallel }_{L{M}_X^{{\phi }_1}}.\hfill \end{array}$$

Thus, the proof is concluded.  □

Example 5.

Interestingly, when $X=L^p\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), Theorem 5 reduces to the result of [3]; when $X=L^{p(·)}\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), it reduces to the result of [26].

Theorem 6.

Consider the following conditions:

• $T_b$ is the commutator in (10) satisfying (6);
• X is a ball Banach function space with M bounded on X and $X^{\prime }$, and $T_b$ bounded on X;
• $b\in {\mathrm{BMO}}_X\left({\mathbb{R}}^n\right)$;
• ${\phi }_1,{\phi }_2:{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$ obey (12);

• the commutator $T_b$ maps ${\widetilde{LM}}_X^{{\phi }_1}\left({\mathbb{R}}^n\right)$ boundedly into ${\widetilde{LM}}_X^{{\phi }_2}\left({\mathbb{R}}^n\right)$, satisfying

  $$\parallel T_b{f\parallel }_{{\widetilde{LM}}_X^{{\phi }_2}}\lesssim {\parallel f\parallel }_{{\widetilde{LM}}_X^{{\phi }_1}},f\in {\widetilde{LM}}_X^{{\phi }_1}\left({\mathbb{R}}^n\right).$$

Proof. 

The proof can be given via adapting the argument of Theorem 4 and utilizing the definition of ${\widetilde{LM}}_X^{\phi }$.  □

Example 6.

Interestingly, when $X=L^p\left({\mathbb{R}}^n\right)$ ($1<p<\infty$), Theorem 6 reduces to the result of [3].

5. Regularity for Solutions to Partial Differential Equations

We now turn to the proof of the regularity estimates for solutions of (1). It suffices to consider the case $q\ge 2$, as the case $1<q<2$ will be recovered thereafter by duality. We begin with the elliptic equation

$$\mathcal{L}u=divf\left(x\right)\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega },$$

where the coefficient functions $a_{ij}$ satisfy

$$a_{ij}\left(x\right)\in L^{\infty }\left(\mathrm{\Omega }\right)\cap VMO,\forall i,j=1,\dots ,n,\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega };$$

(13)

$$a_{ij}\left(x\right)=a_{ji}\left(x\right),\forall i,j=1,\dots ,n,\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega };$$

(14)

$$\exists \tau >0:{\tau }^{-1}{\left|\xi \right|}^2\le a_{ij}\left(x\right){\xi }_i{\xi }_j\le \tau {\left|\xi \right|}^2\forall \xi \in {\mathbb{R}}^n,\mathrm{a}.\mathrm{e}.x\in \mathrm{\Omega }.$$

(15)

To derive the first derivatives of solutions for interior and boundary regularity analysis in generalized local Morrey spaces built upon ball Banach function spaces, we employ an explicit representation formula

$$\mathrm{\Gamma }(x,\xi )={\displaystyle \frac{1}{n(2-n){\omega }_n\sqrt{det\left(a_{ij}\left(x\right)\right)}}}{\left(\sum _{i,j=1}^n{A}_{ij}\left(x\right){\xi }_i{\xi }_j\right)}^{(2-n)/2}$$

for a.e. x and all $\xi \in {\mathbb{R}}^n\backslash \left\{0\right\}$, where $A_{ij}$ are denoted as the entries of the inverse of the matrix ${\left(a_{ij}\right)}_{i,j=1,\dots ,n}$,

$${\mathrm{\Gamma }}_i(x,\xi )={\displaystyle \frac{\partial }{\partial {\xi }_i}}\mathrm{\Gamma }(x,\xi ),{\mathrm{\Gamma }}_{ij}(x,\xi )={\displaystyle \frac{\partial }{\partial {\xi }_i\partial {\xi }_j}}\mathrm{\Gamma }(x,\xi )$$

and

$$M=\underset{i,j=1,\dots ,n}{max}\underset{\left|\alpha \right|\le 2n}{max}{∥{\displaystyle \frac{{\partial }^{\alpha }{\mathrm{\Gamma }}_{ij}(x,\xi )}{\partial {\xi }^{\alpha }}}∥}_{L^{\infty }(\mathrm{\Omega }\times {\mathbb{R}}^n)}.$$

In the variable $\xi$, the functions ${\mathrm{\Gamma }}_{ij}(x,\xi )$ qualify as Calderón–Zygmund kernels according to [27].

Lemma 6.

Let $a_{ij}\left(x\right),i,j=1,\dots ,n$ satisfy (13)–(15) and $B_r$ be some ball such that $B_r\subset \mathrm{\Omega }$, where $B_r$ stands for a ball with radius r. Let u be a solution of $\mathcal{L}u=\mathrm{div}f$ and let $\varphi \in C_0^{\infty }\left(B_r\right)$ be a standard cut-off function:

$$\varphi \left(x\right)=1\mathit{in}{B}_{\sigma r}\mathrm{with}0<\sigma <1,\varphi \left(x\right)=0\mathit{for}\mathit{every}x\notin B_r.$$

Also let

$$\mathcal{L}\left(\varphi u\right)=\mathrm{div}G+g,$$

where G and g are defined as follows:

$$G=\varphi f+uAD\varphi andg=(ADu,D\varphi )-(f,D\varphi ).$$

Then, if $v=\varphi u$, for every $i=1,\dots ,n$, the representation formula is as follows:

$$\begin{array}{cc}\hfill {\partial }_{x_i}v\left(x\right)& =\mathrm{P}.\mathrm{V}.{\int }_{B_r}{\mathrm{\Gamma }}_i(x,x-y)\left\{(a_{ij}\left(x\right)-a_{ij}\left(y\right)){\partial }_{y_j}v\left(y\right)-G_j\left(y\right)\right\}dy\hfill \\ & +c_{ij}{G}_j\left(x\right)-{\int }_{B_r}{\mathrm{\Gamma }}_i(x,x-y)g\left(y\right)dy\forall x\in B_r\hfill \end{array}$$

where $c_{ij}={\int }_{\left|\xi \right|=1}{\mathrm{\Gamma }}_i(x,\xi ){\xi }_{j}d{r}_{\xi }$.

The proof of our main theorem is as follows:

Theorem 7.

Under the assumptions of Lemma 6, suppose condition (4) is fulfilled and ${\phi }_1\lesssim {\phi }_1$. Let u be a solution of (1) satisfying ${\partial }_{x_i}u\in L{M}_X^{{\phi }_1}\left(\mathrm{\Omega }\right)$ and $f\in {\left[L{M}_X^{{\phi }_1}\left(\mathrm{\Omega }\right)\right]}^n$ for all $i=1,\dots ,n$. Then, for any compact set $E\subset \mathrm{\Omega }$ and any standard cut-off function $\phi \in C^{\infty }\left(\mathrm{\Omega }\right)$, the following holds:

(i) 

${\partial }_{x_i}u\in L{M}_X^{{\phi }_2}\left(E\right)$ for all $i=1,\dots ,n$,

(ii) 

$\parallel {\partial }_{x_i}{u\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}\lesssim {\parallel u\parallel }_{L{M}_X^{{\phi }_1}\left(E\right)}+\parallel {\partial }_{x_i}{u\parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}+{\parallel f\parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}$ for all $i=1,\dots ,n$.

Proof. 

Let $E\subset \mathrm{\Omega }$ be an arbitrary compact set. By combining the representation formula from Lemma 6 with the boundedness results established in Theorems 3, 5, and 1, we obtain

$$\begin{array}{cc}\hfill \parallel {\partial }_{x_h}{\left(\varphi u\right)\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}& \lesssim \parallel [a_{ij},\varphi ]{\partial }_{x_h}\left(u\varphi \right){\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}+{\parallel TG\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}+\parallel I_1{g\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}+{\parallel G\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}\hfill \\ \hfill & \lesssim {\parallel a\parallel }_{\ast }\parallel {\partial }_{x_h}{\left(u\varphi \right)\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}+{\parallel G\parallel }_{L{M}_X^{{\phi }_1}\left(E\right)}+{\parallel g\parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}+{\parallel G\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)},\hfill \end{array}$$

where the norm ${\parallel a\parallel }_{\ast }$ is computed over the set B.

The assumption ${\phi }_2\lesssim {\phi }_1$ yields the following norm estimate for ${\parallel f\parallel }_{L{M}_X^{{\phi }_2}}$:

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{L{M}_X^{{\phi }_2}}& =\underset{r>0}{sup}{\phi }_2{(x_0,r)}^{-1}\parallel {\chi }_{B(x_0,r)}{\parallel }_X^{-1}{\parallel {\chi }_{B(x_0,r)}f\parallel }_X\hfill \\ & \lesssim \underset{r>0}{sup}{\phi }_1{(x_0,r)}^{-1}\parallel {\chi }_{B(x_0,r)}{\parallel }_X^{-1}{\parallel {\chi }_{B(x_0,r)}f\parallel }_X\hfill \\ & ={\parallel f\parallel }_{L{M}_X^{{\phi }_1}}.\hfill \end{array}$$

Combining the preceding estimate with the condition ${\parallel a\parallel }_{\ast }<{\displaystyle \frac{1}{2}}$, we conclude that

$$\begin{array}{cc}\hfill \parallel {\partial }_{x_h}& {\left(\varphi u\right)\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}\hfill \\ & \lesssim {\parallel a\parallel }_{\ast }\parallel {\partial }_{x_h}{\left(u\varphi \right)\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}+{\parallel G\parallel }_{L{M}_X^{{\phi }_1}\left(E\right)}+{\parallel g\parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}\hfill \\ & \sim {\parallel a\parallel }_{\ast }\parallel {\partial }_{x_h}{\left(u\varphi \right)\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}{+\parallel \varphi f+uA\nabla \varphi \parallel }_{L{M}_X^{{\phi }_1}\left(E\right)}+{\parallel \langle A\nabla u,\nabla \varphi \rangle -\langle f,\nabla \varphi \rangle \parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}\hfill \\ & \lesssim \parallel {\partial }_{x_h}{\left(u\varphi \right)\parallel }_{L{M}_X^{{\phi }_2}\left(E\right)}+{\parallel f\parallel }_{L{M}_X^{{\phi }_1}\left(E\right)}+{\parallel u\parallel }_{L{M}_X^{{\phi }_1}\left(E\right)}+\parallel {\partial }_{x_i}{u\parallel }_{L{M}_Y^{{\phi }_2}\left(E\right)}+{\parallel f\parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}\hfill \\ & \lesssim {\parallel u\parallel }_{L{M}_X^{{\phi }_1}\left(E\right)}+\parallel {\partial }_{x_i}{u\parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}+{\parallel f\parallel }_{L{M}_Y^{{\phi }_1}\left(E\right)}.\hfill \end{array}$$

We have completed the proof.  □

Theorem 8.

Under the hypotheses of Lemma 6, further assume that condition (4) holds and that ${\phi }_2\lesssim {\phi }_1$. Let u be a solution to (1) satisfying ${\partial }_{x_i}u\in {\tilde{M}}_X^{{\phi }_2}\left(\mathrm{\Omega }\right)$ and $f\in {\left[{\tilde{M}}_X^{{\phi }_1}\left(\mathrm{\Omega }\right)\right]}^n$ for all $i=1,\dots ,n$. Then, for any compact set $E\subset \mathrm{\Omega }$ and any standard cut-off function $\phi \in C^{\infty }\left(\mathrm{\Omega }\right)$, the following regularity estimates hold:

(i) 

${\partial }_{x_i}u\in {\tilde{M}}_X^{{\phi }_2}\left(E\right)$ for all $i=1,\dots ,n$,

(ii) 

$\parallel {\partial }_{x_i}{u\parallel }_{{\tilde{M}}_X^{{\phi }_2}\left(E\right)}\lesssim {\parallel u\parallel }_{{\tilde{M}}_X^{{\phi }_1}\left(E\right)}+\parallel {\partial }_{x_i}{u\parallel }_{{\tilde{M}}_Y^{{\phi }_1}\left(E\right)}+{\parallel f\parallel }_{{\tilde{M}}_Y^{{\phi }_1}\left(E\right)}$ for all $i=1,\dots ,n$.

Proof. 

This theorem can be proved by adapting the argument of Theorem 7 and utilizing the definition of ${\tilde{M}}_X^{\phi }$.  □

Example 7.

Let $X=L^p\left({\mathbb{R}}^n\right)$ and $Y=L^q\left({\mathbb{R}}^n\right)$ with $1<p<q<\infty$ and $1/p-1/q=1/n$. From Theorems 7 and 8, we know that our result partially recovers the result of Guliyev, Omarova, Ragusa, and Scapellato [3].