Regularity results in grand variable exponent Morrey spaces and applications

1.   Introduction

In this paper the boundedness of commutators of Calderón–Zygmund operators in grand variable exponent Morrey spaces ($GVEMSs$ briefly) are studied under the condition that the variable exponents of the spaces satisfy the log–Hölder continuity condition. As a consequence of the general result we have, in particular, the boundedness of the commutator of the Calderón–Zygmund singular integral operator in these spaces defined on a bounded domain in $\mathbb{R}^n$. Based on the latter result, we study the regularity problem for elliptic partial differential equations ($PDE$s briefly).

One-weight estimates for commutators of sublinear operators in constant exponent weighted grand Morrey spaces ($CEWGMSs$ briefly) were investigated in ^[1] (see ^[2] for the unweighted case). The boundedness of operators of harmonic analysis via the Rubio de Francia's extrapolation in $CEWGMS$s with Muckenhoupt weights was studied in ^[3] and ^[4] (see also ^[5] for similar problems beyond the Muckenhoupt classes). The results of ^[3] and ^[4] were applied by the authors to study regularity of elliptic $PDE$s in the frame of $CEWGMS$s. We mention the paper ^[6] for the boundedness of commutators of sublinear operators in generalized Morrey spaces with constant exponent.

The study of function spaces with variable exponents is a very active area of research nowadays. A variable exponent Lebesgue space ($VELS$ briefly) $L^{p(\cdot)}$ is a special case of the space introduced by W. Orlicz in the 1930s and subsequently generalized by I. Musielak and W. Orlicz. These spaces are also called Nakano ^[7] spaces. For mapping properties of operators of harmonic analysis in $VELS$, we refer to the monographs ^[8,9] and the survey ^[10].

Grand Lebesgue spaces $L^{p)}(\Omega)$ defined on bounded domains $\Omega$ were introduced in 1992 by T. Iwaniec and C. Sbordone ^[11] (see ^[12] for further generalization). In subsequent years, quite a number of problems of harmonic analysis and the theory of non-linear differential equations were studied in these spaces (see, e.g., the monograph ^[13] and references cited therein). Grand variable exponent Lebesgue spaces ($GVELS\mathrm{\; s}$ briefly) were introduced in ^[14] (see also ^[15] for more precise spaces).

Morrey spaces describes regularity problems for solutions of elliptic $PDE$s more precisely than Lebesgue spaces. Classical Morrey spaces were introduced by C. B. Morrey ^[16] in 1938 and applied to the regularity problems of solutions of $PDE$s. Variable exponent Morrey spaces appeared in ^[17], while constant exponent grand Morrey spaces ($CEGMS$s briefly) were introduced and studied in ^[5] (see also ^[18] for further generalizations). Grand variable exponent Morrey spaces ($GVEMS$s briefly) were introduced and studied from different viewpoints in the papers ^[19,20,21]. For the progress in the directions of grand function spaces, see, e.g., ^[22].

Finally, we mention that some of the results of this paper were announced in ^[23].

2.   Preliminaries

Let $(X, d, \mu)$ be a quasi-metric measure space ($QMMS$ briefly) with a quasi-metric $d$ and measure $\mu$, i.e., $X$ is a topological space, $\mu$ is a complete measure on the $\sigma$-algebra defined on $X$, and $d$ is a function (quasi-metric) $d: X \times X \rightarrow$ $[0, \infty)$ that satisfies the following conditions:

(a) $d(x, y) = 0$ if and only if $x = y$;

(b) There is a constant $a_0 > 0$ such that $d(x, y) \leq a_0 d(y, x)$ for all $x, y \in X$;

(c) There is a constant $a_1 > 0$ such that $d(x, y) \leq a_1(d(x, z)+d(z, y))$ for all $x, y, z \in X$.

It is also assumed that all balls $B(x, r): = \{y \in X: d(x, y) < r\}$ with center $x$ and radius $r$ in $X$ are measurable, $\mu\{x\} = 0$ for all $x \in X$, and that the class of continuous functions with compact supports is dense in the space of integrable functions on $X$ ($L^1(X)$). We will assume that for all sufficiently small positive $r, \rho$, $0 < r < \rho$,

We say that a measure $\mu$ satisfies the doubling condition if there is a positive constant $C_{dc}$ such that for all $x\in X$ and $r > 0$,

We will deal with the $QMMS$ with doubling measure. Such a $QMMS$ is called a space of homogeneous type ($SHT$ briefly).

A measure $\mu$ satisfies the reverse doubling condition if there are positive constants $\eta_1 > 1$ and $\eta_2 > 1$ such that for all $x\in X$ and sufficiently small $r > 0$,

In this case we write $\mu\in RD_{\eta_1, \eta_2}$.

There are many important examples of an $SHT$:

(a) Carleson (regular) curves on ${\Bbb{C}}$ with arc-length measure $d\nu$ and Euclidean distance on ${\Bbb{C}}$;

(b) Nilpotent Lie groups with Haar measure and homogeneous norm (homogeneous groups);

(c) Bounded domain $\Omega$ in ${\Bbb{R}}^n$ together with induced Lebesgue measure satisfying so called ${\mathcal{A}}$ condition, i.e., there is a positive constant $C$ such that for all $x\in \Omega$ and $\rho \in\left(0, d_{\Omega}\right)$,

where $d_{\Omega}$ is a diameter of $\Omega$ and ${\widetilde{B}}(x, \rho): = \Omega \cap B(x, \rho)$. In this case, balls are the sets ${\widetilde{B}}(x, \rho)$, $x\in \Omega$.

Notation:

By $c$ and $C$ we denote various absolute positive constants, which may have different values even in one and the same line;

$f_B$ denotes the integral average of $f$, i.e., $f_B: = \frac{1}{\mu B} \int_B f \mathrm{\; d} \mu$;

$p^{\prime}(\cdot)$ stands for the conjugate exponent function defined by $1 / p(\cdot)+1 / p^{\prime}(\cdot) = 1$;

$\bar{a}: = a_1\left(a_1\left(a_0+1\right)+1\right)$ with the quasi-metric constants $a_0$ and $a_1;$

$d_X$ denotes the diameter of $X$, i.e., $d_X: = \sup\{ d(x, y): \; x, y \in X \}$;

by the symbol $D(X)$, we denote the class of bounded functions on $X$ with compact supports;

we denote $a B: = B(x, a r)$ for a ball $B: = B(x, r)$, where $a$ is a positive constant;

$B_k\left(x_0, r\right): = \left\{x \in X: d\left(x_0, x\right) < \eta^k r\right\}$ for a constant $\eta > 1$, where $k \in \mathbb{Z}$;

$A_k\left(x_0, r\right): = B_k\left(x_0, r\right) \backslash B_{k-1}\left(x_0, r\right), k \in \mathbb{Z}$, where $x_0$ is a point in $X$;

If $\mu(X) < \infty$, we will assume that $m_0$ is an integer depending on $r > 0$ such that the number $d_{x_0}: = \sup _{x \in X} d\left(x_0, x\right)$ belongs to the interval $\left[\eta^{m_0} r, \eta^{m_0+1} r\right)$, where $\eta$ is a certain positive number.

Throughout the paper we assume that $\mu(X) < \infty$.

2.1.   Grand variable exponent Morrey spaces ($GVEMS$)

We denote by $P(X)$ the family of all real-valued $\mu$-measurable functions $p(\cdot)$ on $X$ such that

where

We say that a variable exponent belongs to the class $\mathcal{P}^{\log }(X)$ if there is a positive constant $\ell$ such that for all $x, y \in X$ with $d(x, y) \leq 1 / 2$,

The variable exponent Lebesgue space, denoted by $L^{p(\cdot)}(X)$, is the linear space of all $\mu$ -measurable functions $f$ on $E$ for which

The norm in $L^{p(\cdot)}(X)$ is defined as follows:

Let $\lambda(x)$ be a measurable function on $X$ with values in $[0, 1]$. Denote by $L^{p(\cdot), \lambda(\cdot)}(X)$ the variable exponent Morrey space ($VEMS$ briefly), which is the class of measurable functions on $X$ such that

The $GVEMS$ denoted by $L^{p(\cdot), \lambda(\cdot), \theta}(X)$, where $\theta > 0$ and $\lambda(\cdot) \in[0, 1]$, are defined with respect to the following norm:

We are interested in the case when $\lambda$ is constant. In particular, in this case, grand variable exponent Morrey space $L^{p(\cdot), \lambda, \theta}(X)$ is defined by the norm:

2.2.   Commutators of Calderón–Zygmund operators

The space of functions of bounded mean oscillation, denoted by $BMO(X)$, is the set of all real-valued locally integrable functions such that

It is well-known that $BMO(X)$ is a Banach space with respect to the norm $\|\cdot\|_{BMO(X)}$ when we regard the space $BMO(X)$ as the class of equivalent functions modulo additive constants.

An equivalent norm for $\|\cdot\|_{BMO(X)}$ is defined as

Let $U$ be an operator defined on some subclass of $\mu-$ measurable functions and let $b$ be a locally integrable function on $X$. We define the commutator $U_b f$ as

Commutators are very useful when studying problems related with regularity of solutions of elliptic partial differential equations of the second order (see, e.g., ^[24]).

We are interested in commutators of Calderón–Zygmund operators defined on an $SHT$ (see ^[25] for the boundedness results of such commutators in weighted classical Lebesgue spaces with Muckenhoupt weights).

Let $K$ be the Calderón–Zygmund operator (see, e.g., ^[26])

where $k$ is the Calderón–Zygmund kernel $k: X \times X \backslash\{(x, x): x \in X\} \rightarrow \mathbf{R}$ satisfying the conditions:

$(i)$ $|k(x, y)| \leq \frac{C}{\mu B(x, d(x, y))}, \quad x, y \in X, \quad x \neq y;$

$(ii)$ $\left|k\left(x_1, y\right)-k\left(x_2, y\right)\right|+\left|k\left(y, x_1\right)-k\left(y, x_2\right)\right| \leq C w\left(\frac{d\left(x_2, x_1\right)}{d\left(x_2, y\right)}\right) \frac{1}{\mu B\left(x_2, d\left(x_2, y\right)\right)},$

for all $x_1, x_2$ and $y$ with $d\left(x_2, y\right) \geq C d\left(x_1, x_2\right)$, where $w$ is a positive nondecreasing function on $(0, \infty)$ which satisfies the $\Delta_2$ condition $w(2 t) \leq c w(t)$ $(t > 0)$ and the Dini condition $\int_0^1 w(t) / t \mathrm{\; d} t < \infty$. It is also assumed that $k$ is such that $K f$ exists almost everywhere on $X$ in the principal value sense for all $f \in L^2(X)$ and that $K$ is bounded in $L^2(X)$.

For the Calderón–Zygmund operator $K$ we will have the following assumptions on their commutators $K_b$:

3.   Boundedness result

Let us recall that by the symbol $D(X)$ is denoted the class of all bounded functions on $X$ with compact supports.

To prove the main boundedness result, we need some auxiliary statements.

Lemma 3.1. (see ^[27]) The following inequality holds for all $b \in BMO(X)$ :

where $A: = D_{dc}^{\log _2 \bar{a}+1}$ with the doubling constant $D_{dc}$.

Some relations between Lebesgue spaces with variable exponent and $BMO$ spaces are given in ^[28]. The proof of ^[28] enables us to formulate it for an $SHT$:

Lemma 3.2. For all $b \in B M O\left(X \right)$ and all $j, i \in \mathbb{Z}$ with $j > i,$ we have that

and

where the constant $C_{p(\cdot)}$ depending on $p(\cdot)$ is such that

for some small positive constant $\delta$.

For the next statement we refer, e.g., to ^[9] (P.9, Lemma 1.7), ^[8] (Lemma 3.3.1):

Lemma 3.3. Let $p, q\in{P} \left(X\right)$ and $q(x) \leqslant p(x)$ almost everywhere, and $\frac{1}{r(x)}: = \frac{1}{q(x)}-\frac{1}{p(x)}$. If $1 \in L^{r (\cdot)}\left(X \right)$, then

The next auxiliary statement for constant exponents was proved in ^[1].

Lemma 3.4. Let $1 < p_{-} \leq p(x) \leq p_{+} < \infty, \theta > 0, 0 < \lambda < 1 / p_+$. Let $p(\cdot)\in {\mathcal{P}}^{\log}(X)$. Suppose that $\sigma$ is a positive constant less than $p_{-}-1$. Let $\eta$ be a positive constant $\eta = \max \{\eta_1, \overline{a} \}$. Then there is a positive constant $C\equiv C_{p(\cdot), BMO}$ such that for all $f \in L^{p(\cdot), \lambda}(X)$, all balls $B: = B\left(x_0, r\right)$, all $\varepsilon \in(0, \sigma)$, and all sufficiently small $r > 0$, the inequality

holds.

Proof. Observe that Hölder's inequality yields

Further, it is easy to see that by adding and subtracting $b_{B_k}$ and $b_{B_{k-1}}$ we find that

This estimate, together with Lemmas 3.1 and 3.2 and condition $p(\cdot)\in {\mathcal{P}}^{\log}(X)$, gives

In the last inequality we used the fact that $\mu$ satisfies the reverse doubling condition $\mu \in RD_{\eta, \eta_2}$ which is guaranteed by the conditions that $\mu$ is doubling and (2.1) is satisfied (see, e.g., ^[29], P.11, Lemma 20 for the details).

Summarizing the estimates above, we obtain the desired result.

□

Proposition 3.1. Let $p(\cdot) \in P(X)\cap \mathcal{P}^{\log }(X)$ and let $\lambda$ be a constant such that $0 < \lambda < 1 / p_{+}$. Let $b \in BMO(X)$. Then there is a positive constant $C$ such that for all $f \in D(X)$,

holds.

Proof. It is known that (see ^[26]) the Calderón–Zygmund operator $K$ is bounded in Morrey space $L^{p(\cdot), \lambda}(X)$ if $p(\cdot) \in P(X)\cap \mathcal{P}^{\log }(X)$. Further, the boundedness of commutators of the Calderón–Zygmund operator in variable exponent Morrey spaces (even in more general spaces) was established in ^[30] in the case of the Euclidean spaces but the proof enables us to conclude that it is true for an $SHT$ with finite measure under the condition that the exponent of the space satisfies the log–Hölder continuity condition. □

Now we formulate the main results of this note:

Theorem 3.1. Let $p(\cdot) \in P(X)\cap \mathcal{P}^{\log }(X)$ and let $\theta$ and $\lambda$ be constants such that $\theta > 0$ and $0 < \lambda < 1 / p_{+}$. Let $b \in BMO(X)$. Then there is a positive constant $C$ independent of $f$ such that

holds for all $f \in D(X)$.

Proof. To prove the theorem, it is enough to prove that

where the norm $\|\cdot\|_{\widetilde{L}^{p(\cdot), \lambda, \theta}(X)}$ is defined by the formula

This follows from the following observation, which holds due to the Hölder inequality in $VELS$s:

Let $\eta: = \max\{\bar{a}, \eta_1 \}$, where $\eta_1$ is the reverse doubling constant. Let us take a ball $B: = B\left(x_0, r\right)$. Using the representation $f = f_1+f_2$, where $f_1 = f \chi_{\eta B}, f_2 = f-f_1$, where $r$ is a sufficiently small positive number, we have

Observe that by Proposition 3.1 we have

To estimate $I_2$, first we observe that if $x \in B$ and $y \notin \eta B$, then

with a positive constant $C$ independent of $x, x_0, y$. Consequently, by condition (2.3), we find that

with the positive constant $C$. Further, by the condition $p(\cdot)\in {\mathcal{P}}^{\log}(X)$, we find that

Observe that Lemma 3.4 yields that

Further, by the condition $p(\cdot) \in {\mathcal{P}}^{\log}(X)$, Lemmas 3.2 and 3.3, we have that

with the positive constant $C_{p(\cdot)}$ independent of $B$ and $\varepsilon$. Consequently, this estimate together with the condition $p(\cdot)\in {\mathcal{P}}^{\log}(X)$ (which implies that $\mu(B)^{1/p_-(B)}\approx \mu(B)^{1/p_+(B)}\approx \mu(B)^{1/p(x_0)}$) and $\mu\in RD_{\eta, \eta_2}$ yield that

Here we used the obvious fact:

Summarizing these estimates we obtain the desired estimate.

□

Corollary 3.1. Let $\Omega$ be a bounded domain in ${R}^n$ such that condition (2.2) is satisfied. Let $K_b$ be the commutator of the Calderón–Zygmund operator defined on $\Omega$. Suppose that $p(\cdot) \in \mathcal{P}^{\log }(\Omega)$ and $\theta$ and $\lambda$ are constants such that $\theta > 0, 0 < \lambda < 1 / p_{+}$. Let $b \in BMO(\Omega, d x)$. Then there is a positive constant $C$ independent of $f$ such that

4.   Applications to $PDE$s

In the last thirty years a number of papers have been devoted to the study of local and global regularity properties of strong solutions to elliptic equations with discontinuous coefficients. To be more precise, let us consider the second-order equation

where $\mathcal{L}$ is a uniformly elliptic operator over the bounded domain $\Omega \subset \mathbb{R}^n, n \geq 2$. We assume that a domain $\Omega$ satisfies $\mathcal{A}$ condition (see (2.2)). In this case $\Omega$, with induced Lebesgue measure and Euclidean metrics is an SHT. Hence, the previous statements are valid for such domains.

Regularizing properties of $\mathcal{L}$ in Hölder spaces (i.e., $\mathcal{L} u \in C^\alpha(\bar{\Omega})$ implies $u \in$ $\left.C^{2+\alpha}(\bar{\Omega})\right)$ have been well studied in the case of Hölder continuous coefficients $a_{i j}(x)$. Also, unique classical solvability of the Dirichlet problem for (4.1) has been derived in this case (we refer to ^[31] and the references therein). In the case of uniformly continuous coefficients $a_{i j}$, an $L^p$-Schauder theory has been elaborated for the operator $\mathcal{L}$ (see ^[31,32]). In particular, $\mathcal{L} u \in L^p(\Omega)$ always implies that the strong solution to (4.1) belongs to the Sobolev space $W^{2, p}(\Omega)$ for each $p \in(1, \infty)$. However, the situation becomes rather difficult if one tries to allow discontinuity at the principal coefficients of $\mathcal{L}$. In general, it is well-known (cf. ^[33]) that arbitrary discontinuity of $a_{i j}$ implies that the $L^p$-theory of $\mathcal{L}$ and the strong solvability of the Dirichlet problem for (4.1) break down. A notable exception to that rule is the two-dimensional case ($\Omega \subset \mathbb{R}^2$). It was shown by G. Talenti that the sole condition on measurability and boundedness of the $a_{i j}$ 's ensures isomorphic properties of $\mathcal{L}$ considered as a mapping from $W^{2, 2}(\Omega) \cap W_0^{1, 2}(\Omega)$ into $L^2(\Omega)$. To handle the multidimensional case ($n \geq 3$) requires that additional properties on $a_{i j}(x)$ should be added to the uniform ellipticity in order to guarantee that $\mathcal{L}$ possesses the regularizing property in Sobolev functional scales. In particular, if $a_{i j}(x) \in W^{1, n}(\Omega)$ (cf. ^[30]), or if the difference between the largest and the smallest eigenvalues of $\left\{a_{i j}(x)\right\}$ is small enough (the Cordes condition), then $\mathcal{L} u \in L^2(\Omega)$ yields that $u \in W^{2, 2}(\Omega)$, and these results can be extended to $W^{2, p}(\Omega)$ for $p \in(2-\varepsilon, 2+\varepsilon)$ with sufficiently small $\varepsilon$.

Later the Sarason class $V M O$ of functions with vanishing mean oscillation was used in the study of local and global Sobolev regularity of the strong solutions to (4.1).

Next, we define the space $B M O$ and then the smallest $V M O$ class, where we consider coefficients $a_{i j}$ and later that one where we consider the known term $f$.

In the sequel, let $\Omega$ be an open bounded set in ${R}^n$.

Definition 4.1. Let $f \in L_{\mathrm{loc}}^1(\Omega)$. We define the integral mean $f_{x, R}$ by

where, as before, $\widetilde{B} (x, R) = \Omega \cap B(x, R)$, $x\in \Omega$, and $|\widetilde{B}(x, R)|$ is the Lebesgue measure of $\widetilde{B}(x, R)$. If we are not interested in specifying which the center is, we just use the notation $f_R$.

We now give the definition of Bounded Mean Oscillation functions $(B M O)$ that appeared at first in the note by F. John and L. Nirenberg ^[34].

Definition 4.2. Let $f \in L_{\mathrm{loc}}^1(\Omega)$. We say that $f$ belongs to $B M O(\Omega)$ if the seminorm $\|f\|_*$ is finite, where

Next, we consider the definition of the space of $VMO$ functions, given at first by D. Sarason ^[35].

Definition 4.3. Let $f \in B M O(\Omega)$ and

where $B \rho$ ranges over the class of the balls of $\mathbb{R}^n$ of radius $\rho$. Further, a function $f \in V M O(\Omega)$ if $\lim _{R \rightarrow 0} \eta(f, R) = 0$.

The Sarason class is then expressed as the subspace of the functions in the John-Nirenberg class whose $B M O$ norm over a ball vanishes as the radius of the balls tends to zero. This property implies a number of good features of $V M O$ functions not shared by general $B M O$ functions; in particular, they can be approximated by smooth functions.

This class of functions was considered by many others. At first, we recall the paper by F. Chiarenza et al. ^[36], where the authors answer a question raised thirty years before by C. Miranda ^[37]. In his paper he considers a linear elliptic equation where the coefficients $a_{i j}$ of the higher-order derivatives are in the class $W^{1, n}(\Omega)$ and asks whether the gradient of the solution is bounded if $p > n$. In ^[36], the authors suppose that $a_{i j} \in V M O$ and prove that $D u$ is Hölder continuous for all $p \in (1, +\infty)$.

Also, it is possible to check that bounded uniformly functions are in $V M O$ as well as functions of fractional Sobolev spaces $W^{\theta, \frac{n}{\theta}}, \theta \in (0, 1)$.

The study of Sobolev regularity of strong solutions of (4.1) was initiated in 1991 with the pioneering work by F. Chiarenza et al. ^[24]. It was obtained that if $a_{i j}(x) \in$ $V M O \cap L^{\infty}(\Omega)$ and $\mathcal{L} u \in L^p(\Omega)$, then $u \in W^{2, p}(\Omega)$ for each value of $p$ in the range $(1, \infty)$. Moreover, well-posedness of the Dirichlet problem for (4.1) in $W^{2, p}(\Omega) \cap W_0^{1, p}(\Omega)$ was proved. As a consequence, Hölder continuity of the strong solution or of its gradient follows if the exponent $p$ is sufficiently large.

Thanks to the fundamental accessibility of these two papers ^[36,38], many other authors have used the $V M O$ class to obtain regularity results for $PDE$s and systems with discontinuous coefficients.

Continuing the study of regularity of $PDE$s, we see that Hölder continuity can be inferred for small $p$ if one has more information on $\mathcal{L} u$, such as its belonging to a suitable Morrey class $L^{p, \lambda}(\Omega)$.

Using these spaces a natural problem arises, namely, to study the regularizing properties of the operator $\mathcal{L}$ in Morrey spaces in the case of $V M O$ principal coefficients. In ^[39], L. Caffarelli proved that each $W^{2, p}$-viscosity solution to (4.1) lies in $C^{1+\alpha}(\Omega)$ if $f(x)$ belongs to the Morrey space $M^{n, n \alpha}(\Omega)$ with $\alpha \in(0, 1)$.

One of the main results of this note is to obtain local regularity, in grand Morrey spaces, for highest-order derivatives of solutions of elliptic non-divergence form with coefficients, which can be discontinuous.

We recall that, in the case of continuous coefficients of the above kind of equation, the results were obtained by S. Agmon et al. ^[32]. Later, discontinuous coefficients were considered by S. Campanato ^[40].

Then this paper can be regarded as a continuation of the study of $L^p$ regularity of solutions of second-order elliptic $PDE$s to the maximum-order derivatives of the solutions to a certain class of linear elliptic equations in nondivergence form with discontinuous coefficients (see also ^[1] for related topics).

Let us consider the second-order differential operator

Here we have adopted the usual summation convention on repeated indices. In the sequel, we need the following regularity and ellipticity assumptions on the coefficients of $\mathcal{L}, \forall i, j = 1, \ldots, n$ :

Set $\eta_{i j}$ for the $V M O$-modulus of the function $a_{i j}(x)$ and let $\eta(r) =$ $\left(\sum_{i, j = 1}^n \eta_{i j}^2\right)^{1 / 2}$. We denote by $\Gamma(x, t)$ the normalized fundamental solution of $\mathcal{L}$, i.e.,

for a.a. $x$ and all $\xi\in{\Bbb{R}}^n\setminus \{0\}$, where $A_{ij} (x)$ stands for the entries of the inverse matrix of the matrix $\{ a_{ij}(x) \}_{i, j = 1, \ldots, n},$ and $\omega_n$ is the measure of the unit ball in ${\Bbb{R}}^n.$ We set also

It is well known that $\Gamma_{ij}(x, \xi)$ are Calderón–Zygmund kernels in the $\xi$ variable.

Corollary 3.1 enables us to formulate the next statement:

Theorem 4.1. Let the coefficients of $\mathcal{L}$ satisfy (4.2) and $p(\cdot) \in \mathcal{P}^{\log }(\Omega), 0 < \lambda <$ $1 / p_{+}, \theta > 0$. Let $\Omega$ be a domain satisfying $\mathcal{A}$ condition (see (2.2)). Then there exist positive constants $c = c(n, \kappa, p(\cdot), \lambda, \theta, M)$ and $\rho_0 = \rho_0(c, n)$ such that for every ball $B_\rho \subset \subset \Omega, \rho < \rho_0$ and every $u \in W_0^{2, p}\left(B_\rho\right)$ such that $D_{i j} u \in L^{p(\cdot), \lambda, \theta}\left(B_\rho\right)$, {it holds true that

5.   Conclusions

The authors obtained regularity results for solutions of second-order $PDE$s having discontinuous coefficients in the framework of grand variable exponent Morrey spaces. In the future it will be possible to extend the obtained properties to other kinds of equations, making use of boundedness properties in grand variable exponent Morrey spaces that are proved in the present paper.

Use of AI tools declaration

The authors declare that Artificial Intelligence (AI) tools played no part in the creation of this article.

Acknowledgements

The third author would like to thank the Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam, for the opportunity to work there. The third author is a member of INdAM (Istituto Nazionale di Alta Matematica "Francesco Severi") research group GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni). The authors are grafetul to the referees for carefully reading this manuscript and for their valuable remarks.

Conflict of interest

Maria Alessandra Ragusa is an editorial board member for Electronic Research Archive and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.