Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form Lu:=∑i,j=1naij(x)uxixj=∇·f,for almost allx∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$\end{document}where the coefficients aij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{ij}$$\end{document} belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces LM~{x0}p,φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$\end{document}. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.


Introduction and mathematical background
In this note we consider the following divergence form elliptic equation in a bounded set ⊂ R n , n ≥ 3.
Extended author information available on the last page of the article We assume that L is a linear elliptic operator and its coefficients belong to the space V M O and the vectorial field f = ( f 1 , f 2 , . . . , f n ) is such that f i ∈ L M p,ϕ for i = 1, . . . , n, with 1 < p < ∞ and ϕ positive and measurable function. The space VMO was introduced by Sarason and it is the proper subspace of the John-Nirenberg space BMO whose BMO norm over a ball vanishes as the radius of the ball tends to zero.
In the last few years have been studied several differential problems on nonstandard function spaces (see for instance [21][22][23]) and, in particular, several results have been obtained on Generalized Morrey Spaces (see, for instance, [12]).
Recently, in [5,27,28] the authors studied some regularity results for solutions of linear partial differential equations with discontinuous coefficients in nondivergence form.
Our main result in this paper is the study of local regularity in the Generalized Morrey Spaces L M p,ϕ of the first derivatives of the solutions of the equation under consideration as in the past has been done in L p −spaces and in L p,λ −spaces.
See, for instance, [2] where the author obtains local regularity in the classical Lebesgue spaces L p for the first derivatives of the solutions of the equation with discontinuous coefficients. See, also, [24] in which has been done the same in the Morrey spaces L p,λ . Hearth of the technique is the use of an integral representation formula for the first derivatives of the solutions of Equation (1.1) and the boundedness in L p,ϕ of some integral operators and commutators appearing in this formula.
Precisely, in this work we apply the boundedness on Generalized local Morrey Spaces of singular integral operators and its commutators obtained in [13]. We would like to point out that in the last decades a lot of authors studied the boundedness of such operators in several functional spaces (see e.g. [1,4,14]).
Throughout the paper, we set d = sup x,y∈ |x − y|, B(x, r ) = {y ∈ R n : |x − y| < r } and (x, r ) = ∩ B(x, r ). Furthermore, by A B we mean that A ≤ C B with some positive constant C independent of appropriate quantities. If A B and B A, we write A ≈ B and say that A and B are equivalent.
Let be an open bounded subset of R n , with n ≥ 3, and f be a locally integrable function on . We say that f belongs to the John-Nirenberg space BMO of the functions with bounded mean oscillation if where B ranges in the set of the balls contained in and f B is the integral average of f over B, namely We say that the number f * is the BMO-norm of f .
where B ρ stands for a ball with radius ρ less than or equal to r . The function η(r ) is called VMO-modulus of f . We say that f ∈ B M O is in the space VMO of functions with vanishing mean oscillation if In the sequel we denote η i j the VMO-modulus of the coefficient a i j and For further details on the V M O space, we refer the reader to [25] and to the references therein.
The definition of local B M O space is as follows.
We set In [16], Lu and Yang introduced the central can be regarded as a local version of B M O(R n ), the space of bounded mean oscillation, at the origin. But, they have quite different properties. The classical John-Nirenberg inequality shows that functions in B M O(R n ) are locally exponentially integrable. This implies that, for any 1 ≤ q < ∞, the functions in B M O(R n ) can be described by means of the condition: where B denotes an arbitrary ball in R n . However, the space where C > 0 is independent of b, r 1 and r 2 . We The following condition is essential to the proof of the main result of the paper: A function b is said to satisfy the well known mean value inequality if there exists a constant C > 0 such that for any ball Also, we recall the definition of the classical Morrey Spaces, formulated by Morrey in 1938 in [19].
According to this definition we obtain, for 0 ≤ λ < n, the Morrey space L p,λ under the choice ϕ(x, r ) = r λ−n p : In this note we are interested in the study of regularity properties of solutions to elliptic equations in the local version of Generalized Morrey Spaces. In order to achieve this purpose we need the following definitions.
According to this definition we obtain, for λ ≥ 0, the local Morrey Space L M
Let be a bounded open set in R n , n ≥ 3, let us consider and, fixed x 0 ∈ R n , we suppose that there exists p ∈]1, +∞[ and a positive measurable function ϕ defined on R n × (0, ∞) such that: We say that a function u is a solution of (1 . . , n and for some 1 < p < ∞ and

Calderón-Zygmund kernel and preliminary results
In order to present the representation formula for the first derivatives of a solution of 1.3, we find it convenient to present the definition of Calderón-Zygmund kernel: Many authors obtained several boundedness results for integral operators involving Calderón-Zygmund kernels. For instance, in [3] the authors studied the boundedness of Calderón-Zygmund singular integral operators and commutators on Morrey Spaces. Recently, in [13] the authors extended the previous results in Generalized Local Morrey Spaces.
The previous theorem was proved using the following important result contained in [10].
where C does not depend on r . Then where c does not depend on x 0 and f and for q = 1 where c does not depend on x 0 and f .
Precisely, using the boundedness of the Calderón-Zygmund singular integral oper- [10]), the following theorem is valid that will be crucial in the sequel.

be a Calderón-Zygmund singular integral operator and the measurable function
where C does not depend on r and x 0 .
To prove Theorem 2.3, we first give some auxiliary lemmas. In this section we are going to use the following statement on the boundedness of the weighted Hardy operator where w is a fixed function non-negative and measurable on (0, d).

Lemma 2.4
Let v 1 , v 2 and w be positive almost everywhere and measurable functions on (0, d). The inequality holds for some C > 0 for all non-negative and non-decreasing g on (0, d) if and only if Moreover, if C * is the minimal value of C in (2.3), then C * = B.
and K be a Calderón-Zygmund singular integral operator. Then the inequality We get From the boundedness of K on L p (R n ), (1.2) and Lemma 1.2 (see [29] [inequality (1.3)]) it follows that: From (1.2) and Lemma 1.2 (see [29] [inequality (1. 3)]) for J 2 we have For J 3 , it is known that x ∈ B, y ∈ (2B), which implies 1 By Fubini's theorem and applying Hölder inequality we have Hence, from Lemma 1.2 we get For x ∈ B by Fubini's theorem applying Hölder inequality and from Lemma 1.2 we have

Remark 2.7
The statement of Theorem 2.3 follows by Lemmas 2.4 and 2.6.
In order to achieve the regularity results, we must prove the following theorem.

5)
then for any r ∈ (0, d) the inequality holds with constant c > 0 independent of g, x 0 and r .
By Fubini's theorem, we get that Applying Hölder's inequality, we obtain Thus the inequality T g 2 L p ( (x 0 ,r )) r n p d r s − n p −1 g L q ( (x 0 ,s)) ds + r n p g L q ( ) (2.11) holds for all r ∈ (0, d/2) for q ≥ 1. Finally, combining (2.10) and (2.11), we obtain that holds for all r ∈ (0, d/2) with a constant independent of f , x 0 and r . Let now r ∈ [d/2, d). Then, using (L q ( ), L p ( ))-boundedness of T , we obtain and inequality (2.6) holds.
If r ∈ [d/2, d), then, using the boundedness of T from L 1 ( ) to W L p ( ), we obtain that and, inequality (2.7) holds.
In order to achieve the regularity results, we must prove the following theorem.

Theorem 2.9
Let be an open bounded subset of R n , x 0 ∈ , 1 ≤ q < p < ∞, 1 q = 1 p + 1 n . Let also ϕ 1 (x, r ) and ϕ 2 (x, r ) two positive measurable functions defined on × (0, d) such that the following condition is fulfilled: From Theorem 2.9 we get the following corollary.

Corollary 2.10
Let be an open bounded subset of R n , 1 ≤ q < p < ∞, 1 q = 1 p + 1 n . Let also ϕ 1 (x, r ) and ϕ 2 (x, r ) two positive measurable functions defined on ×(0, d) such that the following condition is fulfilled: where C does not depend on x and r . Then, in the case q > 1 for every g ∈ M q,ϕ 2 ( ), the function T g(x) is a.e. defined, T g belongs to the space M p,ϕ 1 ( ) and there exists c = c(q, ϕ 1 , ϕ 2 , n) > 0 such that T g M p,ϕ 1 ( ) ≤ c g M q,ϕ 2 ( ) .
In the case q = 1 the function T g belongs to the space W M p,ϕ 1 ( ) and there exists c = c(ϕ 1 , ϕ 2 , n) > 0 such that T g W M p,ϕ 1 ( ) ≤ c g M 1,ϕ 2 ( ) .

Application to partial differential equations
Let us consider the divergence form elliptic equation (1.3), in a bounded set ⊂ R n , n ≥ 3. We set , for a.a. x ∈ B and ∀t ∈ R n \{0}, where A i j denote the entries of the inverse matrix of the matrix {a i j (x)} i, j=1,...,n , and ω n is the measure of the unit ball in R n . It is well known that i j (x, t) are Calderón-Zygmund kernels in the t variable.
Let r , R ∈ R + , r < R and ϕ ∈ C ∞ ( ) be a standard cut-off function such that for every ball B R ⊂ , Then if u is a solution of (1.3) and v = ϕu we have where Using the notations above, we are able to recall an integral representation formula for the first derivatives of a solution u of (1.3).
Using the representation formula stated in Lemma 3. {x 0 } ( )] n , x 0 ∈ . Let ϕ ∈ C ∞ ( ) a standard cut-off function. Then, for any K ⊂ compact there exists a constant c(n, p, ϕ 1 , ϕ 2 , dist(K , ∂ )) such that Proof Let K ⊂ be a compact set. Using Lemma and the boundedness of the commutator proved in [13], we obtain the following estimate: where the norm a CV Now we apply the hypothesis ϕ 2 ϕ 1 , obtaining the following estimate for the norm Using the previous estimate we finally obtain that  where 1 p = 1 q + 1 n . In the case ϕ 1 (x, r ) = ϕ 2 (x, r ) we get the following corollaries.