Local Muckenhoupt class for variable exponents

We define $A_{p(\cdot)}^{\rm loc}$ and show that the weighted inequality for local Hardy--Littlewood maximal operator on the Lebesgue spaces with variable exponent. This work will extend the theory of Rychkov, who developed the theory of $A_p^{\rm loc}$ weights. It will also extend the work by Cruz-Uribe. SFO, Fiorenza and Neugebaucer, who considered the Muckenhoupt class for Lebesgue spaces with variable exponents. Due to the setting of variable exponents, a new method of extension of weights will be needed; the extension method is different from the one by Rychkov. A passage to the vector-valued inequality is also done by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. We develop the theory of extrapolation adapted to our class of weights.


Introduction
This paper develops the theory of local Muckenhoupt weights in a variable exponent setting. It mixes the results obtained in [2,3,14]. Also see the textbook [6]. Due to the setting of variable exponents, we cannot directly use the ideas of Cruz-Uribe, Diening, and Hästö [2], Cruz-Uribe SFO, Fiorenza and Neugebauer [3], or Rychkov [14].
Herein, we use the following notation of variable exponents: Let p(·) : R n → [1, ∞) be a measurable function, and let w be a weight. In other words, w : R n → [1, ∞) is a measurable function that is positive almost everywhere. Then the weighted variable Lebesgue space L p(·) (w) collects all measurable functions f such that for some λ > 0. For f ∈ L p(·) (w), the norm is defined by f L p(·) (w) ≡ inf λ > 0 : If w ≡ 1, then · L p(·) (1) = · p(·) and L p(·) (1) = L p(·) (R n ). Thus, we have the ordinary variable Lebesgue space L p(·) (R n ).
The definition of L p(·) (w) slightly differs from that in [3], where the authors considered the theory of Muckenhoupt weights for the Hardy-Littlewood maximal operator M for Lebesgue spaces with variable exponents. Recall that Rychkov established the theory of the local Muckenhoupt class [14]. Here and below, Q denotes the set of all cubes whose edges are parallel to the coordinate axes. Herein we mix the notions considered in [3,14] to define the local Muckenhoupt class with variable exponents. ≡ sup Q∈Q,|Q|≤1 |Q| -1 χ Q L p(·) (w) χ Q L p (·) (σ ) < ∞, where σ ≡ w -1 p(·)-1 and the supremum is taken over all cubes Q ∈ Q with |Q| ≤ 1. Given a cube Q, analogously define A loc p(·) (Q) by restricting the cubes R to those contained in Q.
Remark that if p(x) = 1 for some x ∈ R n , then we define σ (x) = 1.
If p(·) ≡ p is a constant exponent, then A loc p(·) coincides with the class A loc p defined in [14]. Using a different method, we seek to establish that the local analog of the result in [2,3] is available: Let f be a measurable function and let M loc be the local maximal operator given by Needless to say, this is an analog of the Hardy-Littlewood maximal operator given by For the boundedness of M, we postulate the following two conditions on p(·): (1) The local log-Hölder continuity condition, which is given by (1.1) (2) The log-Hölder continuity condition at infinity. That is, there exists p ∞ ∈ [0, ∞) such that Keeping these in mind, we state the main result of this paper.

Theorem 1.2 Let p(·)
: R n → [1, ∞) satisfy conditions (1.1) and (1.2) and 1 < p -≡ essinf x∈R n p(x) ≤ p + ≡ esssup x∈R n p(x) < ∞. For any given w ∈ A loc p(·) , there exists a constant C > 0 such that for all measurable functions f , It is easy to show that w ∈ A loc p(·) is necessary for the boundedness of M loc , since M loc f (x) ≥ 1 |Q| Q |f (y)| dx for all cubes Q with a volume less than or equal to 1 containing x. Additionally, the matters are reduced to the estimate of the following maximal function. We consider the local maximal operator given by for a measurable function f . In fact, if we denote the seven-fold composition of M loc 6 -1 by (M loc 6 -1 ) 7 , then there exists a constant C > 0 such that M loc f ≤ C(M loc 6 -1 ) 7 f for any measurable function f .
Before we go further, we offer some words on the technique of the proof. At first glance, the proof of Theorem 1.2 seems to be a reexamination of the original theorem [3, Theorem 1.5], which is recalled below.
for all measurable functions f . If, in addition, p -> 1, then for all measurable functions f . However, as the example of w(x) = exp(|x|) ∈ A loc p(·) \ A p(·) shows, inequality which is used in the proof of [3, Theorem 1.5], fails for local Muckenhoupt class with variable exponents. Hence, the proof of [3] cannot be used naively for the local Muckenhoupt class. This observation led to the technique of Rychkov, who detailed a method for creating global weights from given local weights. Next, we consider why the technique employed by Rychkov [14] does not work directly. For simplicity, we work in R. In [14], Rychkov considered a symmetric extension of weights. More precisely, given an interval I and a weight w on I, Rychkov defined a weight w I on an interval J adjacent to I mirror-symmetrically with respect to the contact point in I ∩ J. Here, we tried to repeat this procedure to define a weight w I on R. Hence, this method is not applicable because we cannot extend the variable exponents mirrorsymmetrically. For example, if the weight w satisfies w(t) = |t| -1 3 on (-2, 2) and the exponent p(·) satisfies p(t) = 2 on (-1, 1) and p(t) = 4 3 on (3,5), then the weight w (-2,2) defined mirror-symmetrically from w| (-2,2) does not satisfy σ = w (-2,2) -1 p(·)-1 ∈ L 1 loc (2, 6).
Herein, a dyadic grid is the family D a ≡ k∈Z D k,a for a ∈ {0, 1, 2} n . It is noteworthy that for any cube Q there exists R ∈ a∈{0,1,2} n D a such that Q ⊂ R and |R| ≤ 6 n |Q|. As in [11], we reduced the matters to the local maximal operator generated by a family D given by for a measurable function f and a dyadic grid D ∈ {D a : a ∈ {0, 1, 2} n }. In fact, we have Here and below, due to the similarity, we suppose a = (1, 1, . . . , 1). We abbreviate D (1,1,...,1) to D. The other values of a can be handled similarly.
Since we reduced the matters to a dyadic grid D = D (1,1,...,1) , it is natural to define the class A loc p(·) (D).
In an analogy to Theorem 1.2, we can prove the following theorem: then, for any w ∈ A loc p(·) (D), there exists a constant C > 0 such that for any f ∈ L p(·) (w).
The second device is a new local/global strategy. To prove Theorem 1.5 when dealing with local dyadic Muckenhoupt weights, we consider (global) dyadic Muckenhoupt weights. Definition 1.6 Given an exponent p(·) : R n → [1, ∞) with p -> 1 and a weight w, we say that w ∈ A p(·) (D) if where p (·) is the conjugate exponent of p(·), σ ≡ w -1 p(·)-1 and the supremum is taken over all cubes Q ∈ D.
Here, we propose a new method to create a globally regular weight w Q ∈ A p(·) (D), Q ∈ D from a weight w ∈ A loc p(·) (D) in Proposition 2.10. As we will see, this technique is valid only for the dyadic maximal operator (see Remark 2.11). In addition to the local/global strategy which differs from that in Rychkov [14], we use the localization principle due to Hästo [8,Theorem 2.4]. In analogy to Theorem 1.2, we can prove the following theorem: for any measurable function f .
As explained above, Theorem 1.2 will be proven once we prove Theorem 1.5, whose proof uses Theorem 1.7. We note that unlike the proof of Theorem 1.2, that for Theorem 1.7 is an analog of [3, Theorem 1.5]. However, we include its whole proof for the sake of completeness.
Finally, as an application of our results, we will prove the Rubio de Francia extrapolation theorem in our setting of weights, which in turn produces the weighted vector-valued maximal inequality. The theory of extrapolation is a powerful tool in harmonic analysis to extend many results starting from a weighted inequality. Cruz-Uribe and Wang [5] and Ho [9] extended the extrapolation theorem to weighted Lebesgue spaces with variable exponents. We will show the extrapolation theorem for A loc p(·) by applying the boundedness of the local maximal operator.
holds for pairs of functions (f , g) contained in some family F of nonnegative measurable functions. Let p(·) satisfy conditions (1.1) and (1.2) as well as 1 < p -≤ p + < ∞. Also let w ∈ A loc p (·) . Then, Rychkov [14,Lemma 2.11] proved the weighted vector-valued inequality for M loc and w ∈ A loc p as an extension of the results in [1]. Proposition 1.9 Let 1 < p < ∞, 1 < q ≤ ∞, and w ∈ A loc p . Then for any sequence of mea- Recall that Cruz-Uribe et al. extended the same result by Anderson and John [1] to variable Lebesgue spaces. Proposition 1.10 Suppose that p(·) satisfies conditions (1.1) and (1.2), as well as 1 < p -≤ p + < ∞, and let w ∈ A p(·) and 1 < q ≤ ∞. Then for any sequence of measurable functions . (1.4) The following theorem is the weighted vector-valued inequality for the local variable weight: Theorem 1.11 Suppose that p(·) satisfies conditions (1.1) and (1.2), as well as 1 < p -≤ p + < ∞, and let w ∈ A loc p(·) and 1 < q ≤ ∞. Then for any sequence of measurable functions . (1.5) Throughout the paper, we use the following notation. The relation A B means A ≤ CB for some constant C > 0, while A B means A ≥ CB for some constant C > 0. The relation A ∼ B means that A B and B A. For a weight w and measurable set E, we define The rest of this paper is organized as follows. Sect. 2 establishes various preliminaries and some notation. Sect. 3 proves Theorem 1.7, while Sect. 4 proves Theorem 1.5, which includes Theorem 1.2. Finally, as an application, Sect. 5 is devoted to the proof of the weighted vector-valued maximal inequality for A loc p(·) .

Preliminaries
We collect some preliminary facts by investigating some elementary properties of the dyadic grids D 0 k,1 and D k, (1,1,...,1) and recalling the definition of variable Lebesgue spaces. Then we consider classes of weights.
Proof By the property of "max", it is clear that Q ∩ U k Q +1 = ∅. Let us prove Q ⊂ U k Q -1 . First, we prove Q does not contain U k Q . To this end, it suffices prove (Q) ≤ (U k Q ). Since we can concentrate on the x 1 -direction, we may assume n = 1. Write Next, we will show that Q ⊂ U k Q -1 . If Q ⊂ U k Q , this is clear. We assume otherwise. Then, the relations (2.1) hold. Since the left relation in (2.1) holds, we have Similarly, since the right relation in (2.1) holds, we have Therefore, we see that Q ⊂ U k Q -1 .

Weighted variable Lebesgue spaces
For any measurable subset ⊂ R n , denote In particular, when = R n , we simply write p + and p -, respectively.
Then for all f ∈ L p(·) (R n ) and all g ∈ L p (·) (R n ), Let us recall some properties for the variable Lebesgue space L p(·) (R n ). (1.1) and (1.2).

Lemma 2.3 ([12, Lemma 2.2]) Suppose that p(·) is a function satisfying
Then given any set and any measurable function f , Remark 2.6 Let Q be a cube. In Lemmas 2.4 and 2.5, let f = w 1 p(·) χ Q to obtain the following equivalence: A direct consequence of (2.4) is the following: The following inequality is a key tool used in this paper. Although [3, Lemma 2.7] considers Borel measures, we consider Lebesgue measures. Here and below denote by L 0 (R n ) the set of all measurable functions defined over R n . Lemma 2.7 ([3, Lemma 2.7]) Let μ be a weighted Lebesgue measure defined on a measurable set G. Given a set G and two exponents s(·) and r(·) such that Then Finally, recall the localization principle due to Hästo.
for all measurable functions f .

Weights
Here and below, we assume that p(·) takes value in (1, ∞) and satisfies conditions (1.1) and (1.2). First, note that for w ∈ A loc p(·) we have, by the definition of A loc p(·) , for Q ∈ Q with |Q| ≤ 1.
Recall an equivalent definition of A ∞ . We refer to [7, Theorem 7.3.3] for its proof.
(2) There exist constants 0 < C 1 , C 2 < 1 such that given any cube Q and any measurable set E ⊂ Q with |E| > C 1 |Q|, then w(E) > C 2 w(Q). If (2) holds, then it can be arranged so that C 1 and C 2 depend only on the A ∞ constant of w.
The next lemma is important in this paper. Rychkov extended a local weight mirrorsymmetrically but a variable exponent cannot be set in this manner. Hence, we propose a different extension.

Proposition 2.10
Suppose that p -> 1. Let w ∈ A loc p(·) (D). Let I ∈ D be a cube with |I| = 1. Define Then w ∈ A p(·) (D) and Proof Arithmetic shows that w ∈ A loc p(·) (D) if and only if σ ∈ A loc p (·) (D). We also note that We need to estimate We distinguish three cases: • Suppose I ∩ Q = ∅. In this case, by virtue of Lemma 2.3, • Next, suppose Q ⊂ I. In this case, since w ∈ A loc p(·) (D), • Finally, suppose Q ⊃ I. In this case, again by virtue of Lemma 2.3 and the fact that w ∈ A loc p(·) (D), as required. Remark 2.11 Let I ≡ (0, 1) and J ≡ (-1, 0). Define w(t) = t -1 2 on I and w(t) = 1 on J. Denote by A 2 (I) the A 2 class over I. Although w ∈ A 2 (I), w is not in A 2 (I ∪ J ∪ {0}).

Lemma 2.12 Suppose w ∈ A p(·) (D). Then for all Q ∈ D and a measurable subset E of Q,
Here for the sake of convenience, we include the proof.
Proof By Hölder's inequality (see Lemma 2.2), we have as required.

Corollary 2.14 Let w ∈ A p(·) (D). Then
Proof Let P j be the jth parent of P. Then By Lemmas 2.4 and 2.12, Thus, since K is sufficiently large, it follows that The second inequality is proven similarly.

Lemma 2.15 Suppose w ∈ A p(·) (D). Let Q ∈ D and E be a measurable subset of Q.
(

|E| |Q|
Next, we consider some estimates related to the dyadic maximal operator. We define A direct consequence of Lemma 2.16 is that Given a weight W and a measurable function f , define The next lemma reflects the geometric property of D.

Lemma 2.17 ([10, Lemma 2.1]) Given a weight W and
We transform Lemma 2.17 as shown below.

Lemma 2.18
Let {Q k j } k∈Z,j∈J k be a sparse collection with the nutshell {E k j } k∈Z,j∈J k , and let 1 < r < ∞. Also let W ∈ A ∞ (D). Then for all nonnegative f ∈ L 0 (R n ), Proof By Lemmas 2.9 and 2.17, as required.

Lemma 2.20
Suppose that w ∈ A p(·) (D). Let S be a disjoint collection of cubes in dyadic grid D. Then Proof Let Q ∈ S. Then either Q ⊃ P l for some l or Q is included in some P l . Since the first possibility can occur only in one cube, we only have to consider the second possibility. For such a cube Q, we let l be the smallest number such that Q ⊂ P l . Then P l-1 and Q never intersect due to the minimality of l, since Q does not contain P l . Thus, there exists uniquely an integer l such that Q ⊂ P l \ P l-1 . Using Lemma 2.19(3) and Corollary 2.14, we estimate The next lemma is used in Sect. 3.2.

Lemma 2.21 Suppose that w ∈
Proof By (2.7), we obtain By Lemma 2.9, Corollary 2.14, and the fact that σ (Q k j ) ≥ 1 for any (k, j) ∈ H 2 , we obtain

Proof of Theorem 1.7
Sect. 3 uses the notation in Lemma 2.21.
Suppose that w ∈ A p(·) (D) and f ∈ L ∞ c (R n ) satisfy f L p(·) (w) < 1. We may assume f ≥ 0 a.e. by replacing f by |f | if necessary. We write We only have to show that for j = 1, 2, We form a sparse decomposition of f 1 and f 2 separately. The estimates of f 1 and f 2 will be established independently. So suppose that there exists a sparse family {Q k j } k∈Z,j∈J k with the nutshell {E k j } k∈Z,j∈J k such that Since the E k j 's are disjoint, we have

Estimate of M D f 1
We use the sparse decomposition of M D f 1 : .
Using Lemma 2.7 for the measure w(x) dx, we have To complete the estimate of I 2,G , we only have to show that the first sum is bounded by a constant. We calculate We note that thanks to Lemma 2.15(1) and the definition of A p(·) (D). Thus, thanks to Lemma 2.7 applied for the weighted measure σ (x) dx. Consequently, I 2,G 1.

Estimate of I 3,G Set
Accordingly, we consider Let (k, j) ∈ H 1 . Let x + ∈ Q k j satisfy p(x + ) = p + (Q k j ). Then x ∈ Q k j . Also see [3, (5.14)]. Consequently, from Lemma 2.7 applied for the measure w(x) dx, we deduce from the definition of f 2 . We calculate w(x) dx + 1.
Since σ (Q k j ) ≤ 1 and p(x) ≤ p + (Q k j ) for x ∈ Q k j , we have Thus we obtain L p (·) (w) = g p 0 L p(·) (w) .
Combining these two estimates gives the desired result.