Some basic properties and fundamental relations for discrete Muckenhoupt and Gehring classes

In this paper, we prove some basic properties of the discrete Muckenhoupt class Ap\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}^{p}$\end{document} and the discrete Gehring class Gq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}^{q}$\end{document}. These properties involve the self-improving properties and the fundamental transitions and inclusions relations between the two classes.


Introduction
The study of the discrete analogues in harmonic analysis became an active field of research in recent years. For example, the study of regularity and boundedness of discrete operators on l p analogues for L p -regularity and higher summability of sequences has been considered by some authors, see for example [3-7, 24, 25, 28] and the references cited therein.
Whereas some results from Euclidean harmonic analysis admit an obvious variant in the discrete setting, others do not. The main challenge in such studies is that there are no general methods to study these questions and the discrete operators may behave differently from their continuous counterparts as is exhibited by the discrete spherical maximal operator [19]. We confine ourselves in proving some basic properties of discrete Muckenhoupt and discrete Gehring classes. For properties and the structure of classical Muckenhoupt and Gehring classes (in integral forms), the relation between them and their applications in mathematical analysis, we refer the reader to the papers [1, 2, 8-18, 20-23, 27, 29] and the references cited therein. Throughout this paper, we assume that 1 < p < ∞ and I stands For a given exponent p > 1, we define the A p -norm of the discrete weight u by the following quantity: [u] A p := sup where the supremum is taken over all n ∈ I. The discrete weight u is said to belong to the discrete Muckenhoupt class A 1 (A) if, for every n ∈ I, for A > 1, holds and we define the A 1 -norm by the following quantity: The discrete weight u is said to belong to the discrete Muckenhoupt class A ∞ (A) if, for every n ∈ I, for A > 1, and we define the A ∞ -norm by the following quantity: where the supremum is taken over all n ∈ I. The discrete positive weight u is said to belong to the discrete Gehring class G q (G) on the interval I ⊂ Z + , for q > 1 and G > 1 (independent of q), if, for every n ∈ I, For a given exponent q > 1, we define the G q -norm by the following quantity: where the supremum is taken over all n ∈ I. The discrete weight u is said to belong to the discrete Gehring class G ∞ (G) if for every n ∈ I for G > 0. The discrete weight is said to belong to the discrete Gehring class G 1 (G) if, for every n ∈ I, where G > 1 and we define the G 1 -norm by the following quantity: where the supremum is taken over all n ∈ I. The objective of this paper is classified as follows: (1) Prove some basic properties of the discrete Muckenhoupt class A p .
(2) Prove some basic properties of the discrete Gehring class G q .
(3) Prove some fundamental relations between A p and G q . (4) Prove some fundamental relations between A ∞ and G q and their norms. The paper is organized as follows: In the next section, we state and prove some basic lemmas that are needed in the proofs of the main results. In Sect. 3, we present properties of the Muckenhoupt class which include the self-improving property. In Sect. 4, we prove the basic properties of the Gehring class which include also the self-improving property. In Sect. 5, we prove the transition and inclusion relations between the two classes which give embedding relations between A p and G q and also relations between A ∞ and G q and their norms.

Basic lemmas
In what follows, all sequences in the statements of the theorems are assumed to be positive sequences defined on I ⊆ Z + and use the conventions 0 · ∞ = 0 and 0/0 = 0 and where 1/p + 1/q = 1, and p, q > 1. This inequality is reversed for 0 < p < 1 and if p < 0 or q < 0. For instance the inequality holds if 0 < p < 1 or p > 1, q = -1/(p -1) < 0. If p = 1 and q = ∞, the Hölder inequality is given by if n k=1 |u(k)| < ∞, and sup n |v(n)| < ∞. Throughout we assume that u : I → R + is positive sequence and define M q u(n) := 1 n n k=1 u q (k) 1/q , for any real number q and any n ∈ I. Note that, for q = 0, the operator M q takes the form In the following lemma, we state the basic property of the operator M q u which is proved directly by applying Jensen's inequality.

Lemma 1 Let u be a positive weight and p and q be real numbers. If p
We recall that the discrete positive weight u is said to belong to the discrete Gehring class G q (G) on the interval I ⊂ Z + , for q > 1 and G > 1, if, for every n ∈ I, It is clear that the Hölder inequality (in terms of M q u) reads The reverse of (7) is given by (6), which in terms of M reads for some constant G > 1. A generalization of inequality (8) for 1 < p ≤ q, which we call the generalized reverse Hölder inequality, is given in terms of M by for some constant G > 1. In [26], the authors proved the following transition properties which gives a transition relation between the class A p and the class G q .
Theorem 2 Let u be a positive weight and p and q be real numbers. Then u ∈ A p for some p if and only if u ∈ G q for some q.
Remark 1 The equivalence in this theorem gives the transition property between the discrete Muckenhoupt and Gehring classes. The main question which is interesting is what the relation is between p and q for which the inclusions A p ⊂ G q and G q ⊂ A p hold, and this remains an open problem.

Some basic properties of Muckenhoupt weights
In this section, we prove some basic properties of Muckenhoupt weights. The first lemma proves the inclusion of the class A p in the class

Lemma 3 Let u be a positive weight and p be a nonnegative real number. If u ∈ A p (A), then the inequality
holds.
Proof Since u ∈ A p (A), then, for all n ∈ I and A > 1, we have By taking the limit as p tends to ∞, then the right hand side, after using the properties of limits and L'Hôpital's rule, becomes which is the desired inequality (10). The proof is complete.

Theorem 4 Let u be a positive weight and p be a nonnegative real number, and p = p/(p -1) be the conjugate of p. Then u ∈ A p if and only if u
Proof From the definition of the class A p , and since 1 - Furthermore, since 1 1-p = 1p and 1 1-p = 1p, we have This is equivalent to which is the desired result. The proof is complete.
In the next theorem, we prove some basic inclusion properties of Muckenhoupt classes.
Theorem 5 Let u be a positive weight and p, q be nonnegative real numbers. Then the following inclusion relations hold: Proof (1) Let u ∈ A 1 , then there exists A > 1 such that, for all n ∈ I, we have for all 1 ≤ k ≤ n. For p > 1, by using (11) we have, for all n ∈ I, Hence u ∈ A p , which implies that (2) Assume that u ∈ A p , then there exists A > 1 such that for all n ∈ I holds. Now, since 1 < p ≤ q, we see that 1 p-1 ≥ 1 q-1 , and then, by using Lemma 1, we have Then, for all n ∈ I, we obtain Conversely, we shall prove the containment by contradiction. That is, we assume that u ∈ A ∞ and assume, on the contrary, that, for all 1 ≤ p < ∞, u / ∈ A p . Then, for all 1 ≤ p < ∞, we see that This contradicts the assumption that u ∈ A ∞ , then u ∈ A ∞ implies that, for some 1 ≤ p < ∞, u ∈ A p and hence Thus From (12) and (13), we obtain A ∞ = 1≤p<∞ A p . Moreover, by applying L'Hôpital's rule and some limit rules, we obtain which is the desired result. Now, assume that u ∈ A 1 . By Property (1), for any p > 1, A 1 ⊂ A p , then Equality does not hold and to prove it is sufficient to provide an example of a weight u satisfies u ∈ p>1 A p \A 1 . For example: for all p > 1, we have u(n) = n α ∈ A p for α > 1 and u(n) / ∈ A 1 . The proof is complete.
Remark 2 In Theorem 5, we were able to prove the containment A 1 ⊂ p>1 A p . In the following, we present some weights, which does not only satisfy the containment A 1 ⊂ p>1 A p but also satisfy the equalities: (i) u(n) = 1 ∈ A 1 and hence u(n) ∈ A p , (ii) u(n) = n α ∈ A 1 for α ≤ 0 and hence u(n) ∈ A p , (iii) u(n) = 1 log(n+1) ∈ A 1 and hence u(n) ∈ A p .
In the following theorem, we discuss the power rule for weights in the Muckenhoupt class. That is, we discuss the cases for α which satisfies the necessity of the statement: u ∈ A p implies that u α ∈ A p .

Theorem 6 Let u be a positive weight, p be a nonnegative real number. Then
( Proof (1) For 0 ≤ α ≤ 1, and u ∈ A p , we have 1 p-1 ≥ α p-1 > 0, and hence by applying Theorem 2 implies that u ∈ G q (G). That is, Also, Theorem 4 implies that u 1/(1-p) ∈ A p , again Theorem 2 implies that u 1/(1-p) ∈ G q 1 (G 1 ). That is, , or equivalently, u q ∈ A p . If q > q 1 , then by using Hölder's inequality we see that the condition u ∈ G q implies that 1 n n k=1 u q 1 (k) By using (14), (16) and (17), we have , that is, u q 1 ∈ A p . This completes our proof.
In the next theorem, we discuss the relation between an A p -weight and the product of two sequences in the A 1 -class of weights.

Theorem 8 u ∈ A p if and only if there exist u
Proof First, we prove that if u ∈ A p , then u = u 1 u 1-p 2 such that u 1 , u 2 ∈ A 1 , or equivalently we prove that A p ⊂ A 1 (A 1 ) 1-p . Assume that u / ∈ A 1 (A 1 ) 1-p , then for all u 1 and u 2 satisfying Conversely, assume that u 1 , u 2 ∈ A 1 , and 1 < p < ∞, then for all n ∈ I, we have The proof is complete. Proof We prove this property by using by Property (3) in Theorem 5. Since it is clear that u ∈ A p , for any p > 1, if and only if u ∈ A ∞ . Now, we have by Theorem 4 that u ∈ A p if and only if u 1-p = u The proof is complete. The next theorem is a self-improving property of weights in the Muckenhoupt class.
Theorem 10 Let u be a positive weight, p be a nonnegative number. If u ∈ A p , p > 1, then u ∈ A p-, for some > 0.
Proof Let u ∈ A p , for p > 1, then, for A > 1 and all n ∈ I, we have By Theorem 4, u -p /p = u 1-p ∈ A p . Also, by Theorem 2, u -p /p ∈ G q for some q, or equivalently By using (22) and (23) This follows on taking p --1 = p/(p q), or equivalently, taking = p-1 q . Then u ∈ A pfor p > 1 and some > 0. The proof is complete.

Some basic properties of Gehring weights
In this section, we prove some basic properties of Gehring weights. In the next lemma, we present the inclusion relation of Gehring classes G q in G 1 -class of weights for all 1 < q < ∞. .
The proof is complete.
In the next theorem, we present some basic inclusion properties of weights in the Gehring class.

Theorem 12
Let u be a positive weight and p and q be real nonnegative numbers such that p, q > 1. The following properties hold: (1) G ∞ ⊂ G q ⊂ G 1 for all 1 < q ≤ ∞.
Proof (1) Assume that u ∈ G ∞ , then, by the definition of G ∞ , there exists 0 < C < ∞ such that, for all n ∈ I, we have for all 1 ≤ k ≤ n. By applying (24) for all 1 < q < ∞, we have That is, u ∈ G q and hence G ∞ ⊂ G q . Now, the inclusion G q ⊂ G 1 is proved in Lemma 11. This is the desired result.
(2) Assume u ∈ G q . Then there exists G > 1 such that, for all n ∈ I, That is, u ∈ G p , which completes the proof of the second case.
Conversely, assume that u ∈ G 1 and assume, on the contrary, for all 1 < q ≤ ∞, that u / ∈ G q .
That is, for all n ∈ I, This contradicts the assumption u ∈ G 1 , which implies that, for some 1 < q ≤ ∞, u ∈ G q , and By (26) and (28), we have G 1 = 1<q≤∞ G q . Furthermore, by applying L'Hôpital's rule and limit rules, we have That is, u -1 ∈ G p -1 . This completes our proof.
In the next theorem, we prove the self-improving property of Gehring classes.

Theorem 14
Let u be a positive weight, q be a nonnegative number. If u ∈ G q for q > 1, then u ∈ G q+ , for some > 0.
Proof Assume that u ∈ G q for q > 1, then 1 n n k=1 u q (k) 1/q ≤ G 1 n n k=1 u(k).
By applying Theorem 2 we get u ∈ A p for some p, and property (2) in Theorem 6 implies that u α ∈ A p for α > 1 the smallest number satisfying u ∈ G α and u 1/(1-p) ∈ G α (clearly, q > α). Again, by applying Theorem 2, then u α ∈ G s for some s. Without loss of generality, we choose the largest s satisfying u α ∈ G s (A). Then, by using the condition u ∈ G α (B), it satisfies That is, u ∈ G αs . The cases q < s and (q > s with q < αs) are the only valid cases of s and q as otherwise there exists q/α > s satisfying the condition u α ∈ G q/α , which contradicts the assumption. Then u ∈ G q+ for some = αsq > 0. The proof is complete.
The next theorem presents the relation between the two classes G 1 and A ∞ .
Hence, from (43)  Thus by taking the supremum over all n ∈ I, we have By taking the limit of both sides of (45) as q tends to ∞, we have The proof is complete.