Properties of a Generalized Class of Weights Satisfying Reverse Hölder’s Inequality

In this paper, we will prove some fundamental properties of the discrete power mean operator MpuðnÞ = ð1/n∑k=1 upðkÞÞ1/p, for n ∈ I ⊆Z+, of order p, where u is a nonnegative discrete weight defined on I ⊆Z+ the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, we will study the structure of the generalized discrete class BpðBÞ of weights that satisfy the reverse Hölder inequalityMqu ≤ BMpu, for positive real numbers p, q, and B such that 0 < p < q and B > 1. For applications, we will prove some self-improving properties of weights from BpðBÞ and derive the self improving properties of the discrete Gehring weights as a special case. The paper ends by a conjecture with an illustrative sharp example.


Introduction
In [1], Muckenhoupt introduced a full characterization of the A p − class of weights in connection with the boundedness of the Hardy-Littlewood maximal operator in the space L p w ðℝ + Þ with a weight w. Another important class of weights, the Gehring class G q , for 1 < q < ∞, was introduced by Gehring [2,3] in connection with local integrability properties of the gradient of quasiconformal mappings. Due to the importance of these two classes in mathematical and harmonic analysis, the structure of them has been studied by several authors, and various results regarding the relation between them and their applications have been established. We refer the reader to the papers  and the references cited therein.
In recent years, the study of the discrete analogues in harmonic analysis becomes an active field of research. For example, the study of regularity and boundedness of discrete operator on l p analogues for L p − regularity, higher summability, and structure of discrete Muckenhoupt and Gehring weights has been considered by some authors, and we refer the reader to the papers [24][25][26][27][28][29][30][31][32][33][34] and the references they are cited.
We confine ourselves, in this paper in proving some new fundamental properties of a generalized discrete space of weights that satisfy reverse Hölder's inequality and prove some self-improving properties. As special cases, we will derive the self-improving properties of the discrete Gehring weights.
In the following, for the sake of completeness, we present the background and the basic definitions that will be used in this paper. Throughout this paper, ℤ + stands for the set of nonnegative integers, i.e., ℤ + = f1, 2, ⋯g. By an interval J, we mean a finite subset of ℤ + consisting of consecutive integers, i.e., J = fa, a + 1, ⋯, a + ng, a,n ∈ ℤ + , and |J | stands for its cardinality. We assume that 1 < p < ∞ and fix an interval I ⊂ ℤ + of the form I = f1, 2, ⋯, Ng, where N a nonnegative integer (or ½1, N ⊂ ℤ + ). A discrete weight v defined on I is a sequence of nonnegative real numbers.
A discrete weight u defined on I ⊆ ℤ + belongs to the discrete Muckenhoupt class A 2 ðAÞ for p > 1 and A > 1, if the inequality where the supremum is taken over all intervals J ⊂ I. Note that by Hölder's inequality ½A p ðuÞ ≥ 1 for all 1 < p < ∞ and the following inclusions are true: For a given exponent q > 1 and a constant K > 1, a discrete nonnegative weight u defined on I belongs to the discrete Gehring class G q ðKÞ (or satisfies the reverse Hölder inequality) if for every subinterval J ⊆ I, we have For a given exponent q > 1, we define the G q − norm by where the supremum is taken over all intervals J ⊆ I and represents the best constant for which the G q − condition holds true independently on the interval J ⊆ I. Note that by Hölder's inequality ½G q ðuÞ ≥ 1 for all 1 < q < ∞ and the following inclusion is true: By the generalized power mean operator M q u of order q ≠ 0 and nonnegative weight u defined on I, we mean an operator of the form In [27], Böttcher and Seybold considered the operator (8) and proved that if where p > 1 and 1/p + 1/q = 1, then there exists a constant δ > 0 and C 1 < ∞ depending only on p and u such that for all ε ∈ 0, δ and all J of the form jJj = 2 r with r ∈ ℕ (the set of natural numbers). In [34], the authors proved that for any nonnegative weight u defined on I. In the present paper, we consider the class B q p ðBÞ of all nonnegative weights u that satisfy the reverse Hölder inequality where the constant B > 1 is independent of p,q, and J and q > p. The smallest constant B independent on the interval J and satisfies the inequality (12) is called the B q p − norm which is given by We say that u is a B When we fix a constant C > 1, the triple of real numbers ðp, q, CÞ defines the B q p − discrete class: and we will refer to C as the B q p − constant of the class. It is immediate to observe that the classes A p and G q are special cases of the discrete class B q p of weights as follows: In this paper, we aim to study the structure of the general class B q p and use the new properties to prove some self-improving properties. The paper is organized as follows: In Section 2, we state and prove some basic lemmas concerning the bounds of the generalized power mean operator M p . In Section 3, we will establish some lower 2 Journal of Function Spaces and upper bounds of the composition of operators by using two special functions ρ p and ρ q (will be defined later) and prove some inclusion properties. For example, we prove that if u ∈ B q p ðBÞ, then M q u ∈ B δ p ðB 1 Þ with exact values of δ and B 1 : In Section 4, we present some applications of the main results and prove the self-improving property of a monotone weights from B q p , i.e., we will prove that if u ∈ B q p ðBÞ, then u ∈ B λ p ðB 1 Þ with exact values of λ and B 1 : For illustrations, we will derive the self improving property of the discrete Gehring weights as special cases. The paper ends by a conjecture with the selfimproving of the Muckenhoupt weights with an illustrative example.

Basic Lemmas
In this section, we state and prove the basic lemmas and establish some properties of the power mean operators that will be used to prove the main results later. We will assume that I ≪ f1, 2, ⋯, Ng is a fixed finite subset of ℤ + , and we recall the power mean operator M q u that we will consider in this paper is given by for any nonnegative weight u : I ⟶ ℝ + and q ∈ ℝ \ f0g and by M p q uðkÞ, we mean that ðM q uðkÞÞ p : For the sake of conventions, we assume that 0 · ∞ = 0 and 0/0 = 0 and ∑ b k=a yðkÞ = 0, whenever a > b, and The product rule in the discrete form is given by where ΔuðkÞ = uðk + 1Þ − uðkÞ: The summation by parts now is given by Lemma 1. Let p < q and p:q ≠ 0, and u : I ⟶ ℝ + is a nonnegative weight. Then, the following hold for all n ∈ I.
Proof. By applying the second relation in (18) which is the desired equations (21). Similarly, by applying the second relation in (18) which is the equality (22). The proof is complete.

Lemma 2.
Assume that u : I → ℝ + be any nonnegative weight and q ∈ ℝ \ f0g. Then, following properties hold: (1) If u is nonincreasing, then M q u is nonincreasing and M q uðnÞ ≥ uðnÞ, for all n∈I (2) If u is nondecreasing, then M q u is nondecreasing and M q uðnÞ ≤ uðnÞ, for all n∈I Proof. (1)). From the definition of M q u and the fact that u is nonincreasing, we get for q = 1 that For the general case when q ≠ 1, we have also for all n ∈ I that From this inequality, we get that Now, by using (27) and the fact that u is nonincreasing, we obtain that 3

Journal of Function Spaces
and thus M q uðnÞ is nonincreasing.
2). From the definition of M q uðnÞ and the fact that uðnÞ is nondecreasing, we have for q = 1 that For the general case when q ≠ 1, we have also for all n ∈ I that From this inequality, we see that Then, by using inequality (31) and the fact that u is nondecreasing and proceeding as in the first case, we obtain that We proceed as in the proof of the nonincreasing case to get that ΔðM q uðnÞÞ ≥ 0, and thus M q uðnÞ is nondecreasing. The proof is complete.
The following lemma will play an important rule in proving the main results.

Lemma 3.
Let α and β be positive numbers and g: I → ℝ + be any nonnegative weight such that Then, for every r, s ∈ 0, N such that r < s, we have that Proof. The left-side of inequality (33) writes and by multiplying both sides by k −α and summating from The left-side of inequality (36) can be written in the form By applying Fubini's Theorem on the right-hand side, we have that By using the inequality, with γ = −α < 0, and we have Journal of Function Spaces By substituting (41) into (36), we have which implies that which is equivalent to where r = m − 1 > 0. This proves the left-side of inequality (34). Now, the right-side of inequality (33) writes and by multiplying both sides by k −β , and summating from k = m > 1 to s, we have The left-side of inequality (46) can be written in the form This implies by applying Fubini's Theorem on the lefthand side that By using the inequality, with γ = −β < 0, and we have Then, (48) becomes By substituting (51) into (46), we have which implies that which is equivalent to where r = m − 1 > 0. This proves the right-side of inequality (34). The proof is complete.

Fundamental Properties of Power Mean Operators
In this section, we will prove some fundamental properties of the generalized power mean operator M p uðnÞ which is given by where p is assumed to be positive for the rest of the paper. In order to prove the main results, we will use the properties of the function of the variable λ ∈ ð−∞,0Þ ∪ p, ∞Þ. It is clear that the function ρ p ðλÞ is continuous and increases from 1 to +∞ on ð− ∞, 0Þ and from 0 to 1 on ½p, ∞Þ and for λ ≠ p, we have that To understand the importance of the function ρ p ðλÞ, we consider the sequence u 0 ðnÞ = n −1/λ . Then, we have We consider the different cases of the power −p/λ + 1. First, assume that 0 < −p/λ + 1 < 1, and by employing the inequality with γ = −p/λ + 1, we have Then, we have Next, we consider the case when −p/λ + 1 > 1, and by employing inequality (39) with γ = 1 − p/λ > 1, we have Then, we have that The meaning of ρ p ðλÞ now arises from the fact that the sequence u 0 ðnÞ = n −1/λ satisfies the equivalence between M p u 0 ðnÞ and the fraction u 0 ðnÞ/ρ p ðλÞ, for all n ∈ I. Let 0 < p < q and define the function S p,q ðλÞ by The function C p,q ðλÞ is continuous and increases on the interval ð−∞, 0Þ and decreases on the interval ðq, ∞Þ with C p,q ððβ,∞ÞÞ = ð1,∞Þ: Therefore, for any B > 1, the equation has exactly two roots: a positive root λ + ∈ ðq,∞Þ and a negative root λ − ∈ð−∞,0Þ. The nonnegative weight v : I ⟶ ℝ + is said to be belong to B q p ðBÞ if v satisfies the reverse Hölder inequality that is, for all n ∈ I, where the constant B > 1 is independent of p,q, and q > p: Now, we are ready to state and prove the main properties of the operator (55) and the composition of different operators with different powers.

Journal of Function Spaces
Proof. By applying the product rule (19) on the term Δ½ðk − 1ÞM p q uðk − 1Þ with u = k − 1 and v = M p q uðk − 1Þ, we obtain that Now, we find the estimate of the second term in (69) and consider two cases of the behavior of the monotone weight u: First, we assume that u is nondecreasing. Then, by Lemma 2, we have that is also nondecreasing and by applying the elementary inequality (59), for γ =p/q < 1, we obtain By combining (71) and (69), we obtain Next, we assume that u is nonincreasing. Then, by Lemma 2, we see that M q uðkÞ is nonincreasing and by employing the inequality (59) again, we have that By combining (69) and (73), we again obtain the inequality (72). Now, by summing (72) from 1 to n, and applying (21), we obtain From the definition of M q u, we see that the first term in (74) is given by Now, we simplify the term By applying reverse Hölder's inequality for p/q < 1 and p/ðp − qÞ, we obtain that

Journal of Function Spaces
By substituting (75) and (77) into (74), dividing by p½M p ðM q uÞðnÞ p and then applying (66), we obtain By setting we see that the inequality (78) can be written in the form This inequality can be written now as or equivalently This means that λ ∈ ð−∞,λ − ∪ λ + , +∞Þ. The properties of ρ p imply that and since we obtain that which is the desired inequality (68). The proof is complete.
Proof. By applying the product rule (19) on the term Δ½ðk − 1ÞM q p uðk − 1Þ, with u = k − 1 and v = M q p uðk − 1Þ, we obtain that First, we assume that u is nonincreasing. Then, by Lemma 2, we have then M p uðkÞ that is nonincreasing. By employing inequality (39) with γ = q/p > 1, we obtain
(2). Assume that λ < 0. By raising (86) to the power λ, we obtain for m < n that By using the monotonicity of (see Theorem 6), we have that