Research Article | Open Access
S. H. Saker, S. S. Rabie, R. P. Agarwal, "Properties of a Generalized Class of Weights Satisfying Reverse Hölder’s Inequality", Journal of Function Spaces, vol. 2021, Article ID 5515042, 16 pages, 2021. https://doi.org/10.1155/2021/5515042
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 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.
In , 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 , 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 , 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