Regularity in Orlicz spaces for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition ∗

The purpose of this paper is to obtain the global regularity in Orlicz spaces for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition.


Introduction
In this paper we consider the following nondivergence elliptic operator where x = (x 1 , x 2 , . . ., x n ) ∈ R n (n ≥ 3), and establish the regularity in Orlicz spaces for (1.1).It will be assumed that the following assumptions on the coefficients of the operator A and the potential V are satisfied (H 1 ) a ij ∈ L ∞ (R n ) and a ij = a ji for all i, j = 1, 2, . . ., n, and there exists a positive constant Λ such that a ij (x)ξ i ξ j ≤ Λ|ξ| 2 for any x ∈ R n and ξ ∈ R n ; (H 2 ) a ij (x) ∈ V M O(R n ), which means that for i, j = 1, 2, . . ., n, a ij (y) − a B ij dy → 0, r → 0 + , where a B ij = |B ρ (x)| −1 Bρ(x) a ij (y)dy; (H 3 ) V ∈ B q for n/2 ≤ q < ∞, which means that V ∈ L q loc (R n ), V ≥ 0, and there exists a positive constant c 1 such that the reverse Hölder inequality Note that when we say In fact, if V ∈ B ∞ , then it implies that V ∈ B q for 1 < q < ∞.Regularity theory for elliptic operators with potentials satisfying a reverse Hölder condition has been studied by many authors (see [4], [9]- [12], [14], [15]).When A is the Laplace operator and V ∈ B q (n/2 ≤ q < ∞), Shen [10] derived L p boundedness for 1 < p ≤ q and showed that the range of p is optimal.If A is the Laplace operator and V ∈ B ∞ , an extension of L p estimates to the global Orlicz estimates was given by Yao [14] with modifying the iteration-covering method introduced by Acerbi and Mingione [1].For a ij ∈ C 1 (R n ) and V ∈ B ∞ , regularity theory in Orlicz spaces for the operators n i,j=1 ∂ x i (a ij ∂ x j ) + V was proved by Yao [15].Recently, under the assumptions (H 1 )-(H 3 ), the global L p (R n ) estimates for L in (1.1) has been deduced by Bramanti et al [4].
In this paper we will establish global estimates in Orlicz spaces for L which extends results in [4] to the case of the general Orlicz spaces.Our approach is based on an iteration-covering lemma (Lemma 3.1), the technique of "S.Agmon's idea"(see [3], p. 124) and an approximation procedure.
The definitions of Yong functions φ, Orlicz spaces and their properties will be described in Section 2.
We now state the main result of this paper.
then there exists a constant C > 0 such that for any µ 1 large enough, we have where the constants α 1 and α 2 appear in Orlicz spaces, see (2.4), C depends only on n, q, Λ, c 1 , α 1 , α 2 and the VMO moduli of the leading coefficients a ij .
The proof of Theorem 1.1 is based on the following result.
Theorem 1.2Under the same assumptions on φ, a ij , V , q, f as in Theorem Then there exists a constant C > 0 such that where C depends only on n, q, Λ, c 1 , a, K and the VMO moduli of a ij .
Note that Theorem 1.2 and Definition 2.9 easily imply the following result by using the monotonicity, convexity of φ, (2.2) and Remark 2.7.
Corollary 1.3Under the same assumptions on φ, a ij , V , q, f as in Theorem 1.1, Then there exists a constant C > 0 such that where C depends only on n, q, Λ, c 1 , a, K and the VMO moduli of a ij .
Remark 1.4 When we take φ(t) = t p , t ≥ 0 for 1 < p < ∞, then (1.4) is reduced to the classical L p estimates (see [4,Theorem 1]).This paper will be organized as follows.In Section 2 some basic facts about Orlicz spaces and Orlicz-Sobolev spaces are recalled.In Section 3 we prove Theorem 1.2 by describing an iteration-covering lemma (Lemma 3.1) and using the EJQTDE, 2013 No. 78, p. 3 results in [4].Section 4 is devoted to the proof of Theorem 1.1.We first assume u ∈ C ∞ 0 (B R 0 /2 ) satisfying (1.2) and prove that (1.3) is valid by using Theorem 1.2 and "S.Agmon's idea"(see [3], p. 124); then we show that the assumption u ∈ C ∞ 0 (B R 0 /2 ) can be removed by an approximation procedure and a covering lemma in [5].Dependence of constants.Throughout this paper, the letter C denotes a positive constant which may vary from line to line.

Preliminaries
We collect here some facts about Orlicz spaces and Orlicz-Sobolev spaces which will be needed in the following.For more properties, we refer the readers to [2] and [8].
We use the following notation: (2. 3) The following result was obtained in [7]. where Definition 2.5 (Orlicz spaces) Given a Young function φ, we define the Orlicz class K φ (R n ) which consists of all the measurable functions g : R n → R satisfying EJQTDE, 2013 No. 78, p. 4 In the Orlicz spaces L φ (R n ), we use the following Luxembourg norm The space L φ (R n ) equipped with the Luxembourg norm [2], pp.266-274).
(ii) If φ ∈ ∆ 2 , then the mean convergence implies the norm convergence.
The norm is defined by where The following definition is analogous to the definition of the space W 2,p V (R n ) introduced by Bramanti, Brandolini, Harboure and Viviani in [4].
Remark 2.10 (see e.g.[13]) Before the proof of Theorem 1.2, some notions and two useful lemmas are given.Let us introduce the notation where ε ∈ (0, 1) is a small enough constant to be determined later.Let Then u λ satisfies Lu λ = f λ .For any ball B in R n , we use the notations The following lemma is just an analogous version of the result given in [15,Lemma 2.2].Here the selection of λ 0 and the condition of V are different from [15].Lemma 3.1 (Iteration-covering lemma) For any λ > 0, there exists a family of disjoint balls and where F is a zero measure set.Moreover, In analogy with [4, Theorem 13], the following lemma holds by using [4, Theorem 2, Theorem 3], and standard techniques involving cutoff functions and the interpolation inequality (see e.g.[6]).Lemma 3.2 Under the assumptions (H 1 )-(H 3 ), for any γ ∈ (1, q], there exists a positive constant C such that for any x i , ρ x i as in Lemma 3.1 and u ∈ C ∞ 0 (R n ), where C depends only on n, γ, q, c 1 , Λ and the VMO moduli of a ij .
4 Proof of Theorem 1.1 By the technique of "S.Agmon's idea"(see [3], p. 124) and Theorem 1.2, we first prove the following lemma.
Lemma 4.1 Under the same assumptions on φ, a ij , V , q, f as in Theorem 1.1, let u ∈ C ∞ 0 (B R 0 /2 ) satisfy the following equation Then for any µ 1 large enough, where the constant C is independent of µ, and R 0 , α 2 are the constants in the proofs of Theorem 1.2 and (2.4), respectively.
EJQTDE, 2013 No. 78, p. 9 Proof Let ξ ∈ C ∞ 0 (−R 0 /2, R 0 /2) be a cutoff function (not identically zero) and set and where µ ≥ 1 will be chosen later, then ũ(z) It is easy to verify that the coefficients matrix of the operator L still satisfies the assumptions (H 1 ) and (H 2 ).Furthermore, in view of (4.2) and ( 4.3) we find that where For the sake of convenience, we use the following notation where Applying Theorem 1.2 to (4.4), If ξ(t) cos( √ µt) > 0, by (2.4) we have EJQTDE, 2013 No. 78, p. 10 This and (4.2) yield Similarly to (4.7) we get Using (2.4), Thus, By choosing µ 1 large enough, we obtain the following we immediately find that The desired estimate (4.1) follows by taking µ 1 large enough.The lemma is proved.
Furthermore, we shall show that the assumption C ∞ 0 (B R 0 /2 ) can be removed.A covering lemma in a locally invariant quasimetric space was proved by Bramanti et al. in [5].Since the Euclidean space R n is a special locally invariant quasimetric space, the covering lemma also holds in R n .For the convenience to readers, we describe it as follows.
Lemma 4.2 For given R 0 and any κ > 1, there exist R 1 ∈ (0, R 0 /2), a positive integer M and a sequence of points where χ B κR 1 (x i ) (y) is the characteristic function of B κR 1 (x i ), that is, the function equal to 1 in B κR 1 (x i ) and 0 in R n \B κR 1 (x i ).
By Definition 2.9, there exists a sequence {u k } of functions in C ∞ 0 (R n ) such that 12) It follows from Remark 2.7 that Let u 0 k = u k ρ 0 .Then using the properties of ρ, the monotonicity, convexity of φ, (4.13), (2.4) and Remark 2.7, we obtain It follows by (H 1 ) and (4.12) that

. 6 ) 5 3
As usual, we denote by B R (x) the open ball in R n of radius R centered at x and B R = B R (0).EJQTDE, 2013 No. 78, p. Proof of Theorem 1.2