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

Saker, S. H.; Rabie, S. S.; Agarwal, R. P.

doi:https://doi.org/10.1155/2021/5515042

Journal of Function Spaces

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Research Article | Open Access

Volume 2021 | Article ID 5515042 | https://doi.org/10.1155/2021/5515042

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

S. H. Saker,^1,2S. S. Rabie,²and R. P. Agarwal³

Academic Editor: Ismat Beg

Received10 Jan 2021

Accepted04 Feb 2021

Published23 Feb 2021

Abstract

In this paper, we will prove some fundamental properties of the discrete power mean operator of order , where is a nonnegative discrete weight defined on 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 of weights that satisfy the reverse Hölder inequality for positive real numbers , , and such that and . For applications, we will prove some self-improving properties of weights from 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.

1. Introduction

In [1], Muckenhoupt introduced a full characterization of the class of weights in connection with the boundedness of the Hardy-Littlewood maximal operator in the space with a weight . Another important class of weights, the Gehring class , for , 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 [1–23] 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 analogues for 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 [24–34] 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, stands for the set of nonnegative integers, i.e., . By an interval , we mean a finite subset of consisting of consecutive integers, i.e., , , and stands for its cardinality. We assume that and fix an interval of the form , where a nonnegative integer (or ). A discrete weight defined on is a sequence of nonnegative real numbers.

A discrete weight defined on belongs to the discrete Muckenhoupt class for and if the inequality holds for every subinterval , where is the cardinality of the set . A discrete weight defined on is said to belong to the discrete Muckenhoupt class for and if the inequality holds for every subinterval interval . For a given exponent we define the norm by the following quantity where the supremum is taken over all intervals . Note that by Hölder’s inequality for all and the following inclusions are true:

For a given exponent and a constant a discrete nonnegative weight defined on belongs to the discrete Gehring class (or satisfies the reverse Hölder inequality) if for every subinterval , we have

For a given exponent we define the norm by where the supremum is taken over all intervals and represents the best constant for which the condition holds true independently on the interval . Note that by Hölder’s inequality for all and the following inclusion is true:

By the generalized power mean operator of order and nonnegative weight defined on , we mean an operator of the form

In [27], Böttcher and Seybold considered the operator (8) and proved that if where and , then there exists a constant and depending only on and such that for all and all of the form with (the set of natural numbers). In [34], the authors proved that for all for all and if then for any nonnegative weight defined on . In the present paper, we consider the class of all nonnegative weights that satisfy the reverse Hölder inequality where the constant is independent of , and and . The smallest constant independent on the interval and satisfies the inequality (12) is called the norm which is given by

We say that is a weight if its norm is finite, i.e.,

When we fix a constant , the triple of real numbers defines the discrete class: and we will refer to as the constant of the class. It is immediate to observe that the classes and are special cases of the discrete class of weights as follows:

In this paper, we aim to study the structure of the general class 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 . In Section 3, we will establish some lower and upper bounds of the composition of operators by using two special functions and (will be defined later) and prove some inclusion properties. For example, we prove that if , then with exact values of and In Section 4, we present some applications of the main results and prove the self-improving property of a monotone weights from i.e., we will prove that if , then with exact values of and 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 self-improving of the Muckenhoupt weights with an illustrative example.

2. 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 is a fixed finite subset of , and we recall the power mean operator that we will consider in this paper is given by for any nonnegative weight and and by , we mean that For the sake of conventions, we assume that and and , whenever , and

The product rule in the discrete form is given by where The summation by parts now is given by

Lemma 1. Let and , and is a nonnegative weight. Then, the following hold for all .

Proof. By applying the second relation in (18) with , we have which is the desired equations (21). Similarly, by applying the second relation in (18) with , we have which is the equality (22). The proof is complete.

Lemma 2. Assume that be any nonnegative weight and . Then, following properties hold: (1)If is nonincreasing, then is nonincreasing and for all (2)If is nondecreasing, then is nondecreasing and for all

Proof. (1)). From the definition of and the fact that is nonincreasing, we get for that For the general case when , we have also for all that From this inequality, we get that Now, by using (27) and the fact that is nonincreasing, we obtain that and thus is nonincreasing.
2). From the definition of and the fact that is nondecreasing, we have for that For the general case when , we have also for all that From this inequality, we see that Then, by using inequality (31) and the fact that 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 , and thus is nondecreasing. The proof is complete.

The following lemma will play an important rule in proving the main results.

Lemma 3. Let and be positive numbers and be any nonnegative weight such that Then, for every , such that we have that

Proof. The left-side of inequality (33) writes and by multiplying both sides by and summating from to , we have 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 , and we have Then, (38) becomes By substituting (41) into (36), we have which implies that which is equivalent to where . This proves the left-side of inequality (34). Now, the right-side of inequality (33) writes and by multiplying both sides by , and summating from to , we have The left-side of inequality (46) can be written in the form This implies by applying Fubini’s Theorem on the left-hand side that By using the inequality, with , and we have Then, (48) becomes By substituting (51) into (46), we have which implies that which is equivalent to where . This proves the right-side of inequality (34). The proof is complete.

3. Fundamental Properties of Power Mean Operators

In this section, we will prove some fundamental properties of the generalized power mean operator which is given by where 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 . It is clear that the function is continuous and increases from to on and from to on and for , we have that

To understand the importance of the function , we consider the sequence . Then, we have

We consider the different cases of the power . First, assume that , and by employing the inequality with , we have

Then, we have

Next, we consider the case when , and by employing inequality (39) with , we have

Then, we have that

The meaning of now arises from the fact that the sequence satisfies the equivalence between and the fraction , for all . Let and define the function by

The function is continuous and increases on the interval and decreases on the interval with Therefore, for any , the equation has exactly two roots: a positive root and a negative root . The nonnegative weight is said to be belong to if satisfies the reverse Hölder inequality that is, for all , where the constant is independent of , and Now, we are ready to state and prove the main properties of the operator (55) and the composition of different operators with different powers.

Theorem 4. Let and be any nonnegative, monotone weight. If for, then where and are the roots of (65).

Proof. By applying the product rule (19) on the term with and , 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 First, we assume that is nondecreasing. Then, by Lemma 2, we have that is also nondecreasing and by applying the elementary inequality (59), for , we obtain By combining (71) and (69), we obtain Next, we assume that is nonincreasing. Then, by Lemma 2, we see that 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 to and applying (21), we obtain From the definition of , we see that the first term in (74) is given by Now, we simplify the term By applying reverse Hölder’s inequality for and , we obtain that By substituting (75) and (77) into (74), dividing by 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 . The properties of imply that and since we obtain that which is the desired inequality (68). The proof is complete.

Theorem 5. Let and be any nonnegative, monotone weight. If for then where and are the roots of (65).

Proof. By applying the product rule (19) on the term , with and , we obtain that First, we assume that is nonincreasing. Then, by Lemma 2, we have then that is nonincreasing. By employing inequality (39) with , we obtain By substituting (88) into (87), we obtain Next, we assume that is nondecreasing. Then, by Lemma 2, we have that is nondecreasing and by applying the inequality (39) with , we get Now, by combining (87) and (90), we obtain again (89). Now, by summing (89) from to and applying (22), we obtain that The first term in (91) is given by Now, we simplify the second term in (91), by applying Hölder’s inequality for and , to obtain By substituting (92) and (94) into (91), dividing by and applying (66), we obtain Inequality (95) now takes the form By setting we see that inequality (96) becomes or equivalently, This means that . The properties of implies that and since we obtain that which is the required inequality (86). The proof is complete.

The assumptions and the conclusions of Theorems 4 and 5 will be used in proving the following theorems.

Theorem 6. Assume that the conditions in Theorems 4 and 5 hold. Then, the compositions are nonincreasing for all , and the compositions are nondecreasing for all .

Proof. Raise the inequality (68) to the power , we get that Setting we see that satisfies the inequality (34) in Lemma 3 with Therefore, we see that and so we have that If raised the last inequality to the power , we see that this inequality is equivalent to the monotonicity of the compositions Analogously, the inequality (86) and the same proof imply the monotonicity of the compositions The proof is complete.

Theorem 7. Let and be any nonnegative weight, and and are the roots of equation (65). (i)If for and and then(ii)If for and , and then

Proof. (i). Since is either positive or negative, we will discuss the two cases:
(1) Assume that . By raising (86) to the power , we obtain for that By using the monotonicity of (see Theorem 6), we have that Since , by summing (116) from to , dividing by , and raising it to the power , we get Since , we have that . By applying (59) with , we have and then So, we have Similarly, since , then and hence . By applying (39) with , we have and then In this case, we have By substituting (120) and (123) into (117), we obtain that which implies that that is the first relation in (112) holds for .
(2). Assume that . By raising (86) to the power , we obtain for that By using the monotonicity of (see Theorem 6), we have that Since , by summing (128) from to , dividing by , and raising it to the power , we get Since , then and by applying (39) with , we have and then Since , we have that Similarly, since and , then and . If we apply inequality (59) to obtain and then Since , we have that If , we apply (39) to obtain and then In this case, we have From the two cases, we conclude that for all , we have By substituting (132) and (139) into (117), we have This gives us that which is the first relation in (112) in the case . Similarly, we can prove the first relation in (112) by using relation (68). Analogously, we prove the two relations in (113) by using the same technique and inequalities (68) and (86). The proof is complete.

4. Self-Improving Properties

In Theorem 7, we proved that the power mean operators and of the weight satisfy the reverse Hölder inequality with some better exponents. However, the fact that the mean or belongs to some class does not imply that the weight itself belongs to . Thus, Theorem 7 does not guarantee the self-improvement of the summability exponents of the weight . But if we additionally assume the condition of the monotonicity of the weight , then we can obtain the following results for self-improving of exponents.

Theorem 8. Let and be any nonnegative weight belongs to for , and and are the roots of equation (65). (1)If is nonincreasing, and , then(2)If is nondecreasing, and , then

Proof. (1). Since is nonincreasing, then Lemma 2 implies that is also nonincreasing and and hence By applying the second relation in (112), we obtain that By applying the left-inequality in (68) and since is nonincreasing (see Lemma 2), we from (144) and (145) that That is, which is the first relation in (142). Similarly, since is nonincreasing, we have and so By applying the first relation in (112), we obtain that By applying the left-inequality in (86), and using (148), (149), and the fact that is nonincreasing (see Lemma 2), we have that That is, which is the second relation in (142).
(2) Since is nondecreasing, then Lemma 2 implies that is nondecreasing and , and we have that By applying the first relation in (113), we obtain that By applying the right-inequality in (86), we have that By combining (152) and (154), we have that that is, which is the second relation in (143). Again, since is nondecreasing, then Lemma 2 implies that is nondecreasing and and so By applying the second relation in (113), we obtain that By applying the right-inequality in (68), we have that By combining (157) and (159), we have that That is, which is the first relation in (143). The proof is complete.

Now, we derive the self-improving properties of the class

Theorem 9. Let and be any nondecreasing weight, add weight after nondecreasing for . Then, for , where is the root of the equation

Proof. Since , then equation (65) becomes which writes which is the desired equation (162), and the constant is obtained from (142) and given by The proof is complete.

Remark 10. Our technique is only applicable in the case when , and it remains an open problem to prove the main results for all and such that and to be able to get the main results of Muckenhoupt weights similar to the Gehring weights This leads to the following conjecture.

Theorem 11. Let such that and be any nonnegative weight that belongs to for , and and are the roots of equation (65). (1)If is nonincreasing, and , then(2)If is nondecreasing, and then

From this theorem, we can obtain the following sharp result.

Theorem 12. Let and be any nonnegative and nondecreasing belong to for . Then, for , where is the root of the equation

Proof. Since , then equation (65) becomes which is written by By applying the transform , we see that is determined from the equation which is the desired equation (167), and the constant is obtained from (143) and given by The proof is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was supported financially by the Academy of Scientific Research and Technology (ASRT), Egypt (Grant No. 6426), and ScienceUP program (ASRT) is the affiliation of this research.

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Copyright © 2021 S. H. Saker et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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