Abstract
In this paper, we establish the sharp constant for -analogus of Hausdorff operators on central -Morrey spaces. As applications, the sharp constants for the -analogus of Hardy operator and its dual operator, the -analogue of Hardy-Littlewood-Pólya operator, and the -analogue of Hilbert operator on central -Morrey spaces are deduced.
1. Introduction
The quantum calculus (-calculus), initially introduced by Jackson [1, 2], has plenty of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions and other sciences quantum theory, mechanics, and the theory of relativity (see [3–7] for the details).
In recent years, many classical inequalities have been extended to the -calculus setting, such as the Hardy -inequalities [8–11], the Hausdorff -inequalities [12, 13], and some other important -integral inequalities [14, 15]. The aim of this paper is to study the sharp constant for Hausdorff operators in the framework of -calculus on central -Morrey spaces. We first recall the definition of Hausdorff operators on Euclidean spaces. For a given locally integrable function defined on , the Hausdorff operator is defined by
The Hausdorff operators were first introduced in [16] with summability of number series. As is well known, Hausdorff operators contain many famous operators such as the Hardy operator, the adjoint Hardy operator, the Cesàro operator, and the Hardy-Littlewood-Pólya operator as special cases.
Nowadays, Hausdorff operators and their variations have been widely studied by many researchers. For instance, Chen et al. [17–20] and Liflyand et al. [21–25] studied the mapping properties of Hausdorff operators on the Euclidean spaces, such as Lebesgue spaces, the Hardy spaces, and Herz-type spaces. In recent years, the sharp constants for Hausdorff operators on various function spaces such as Lebesgue spaces and (central) Morrey spaces have been built (see [26–32] for more details).
On the other hand, Maligranda et al. [8] derived some Hardy -inequalities and obtained the sharp constants, which extends the corresponding results of the classical Hardy inequalities. Subsequently, Guo and Zhao [12] introduced the -analogue of Hausdorff operators and established the Hausdorff -inequalities, which generalized the main results in [8]. One can also consult [13] for the Hausdorff -inequalities for the multivariable setting. Note that the main results of [12, 13] essentially build the sharp constants for -analogue of Hausdorff operators on -Lebesgue spaces. It is natural to ask whether the -analogue of Hausdorff operators can be bounded on central -Morrey spaces (see the definition below).
To this end, we first introduce some basic notations and definitions of -calculus, which are necessary for understanding this paper. Fix a positive number and a function , the -integral of the -Jackson integral of is defined by and the improper -integral of a function is defined by the series provided that the series on the right-hand sides of (2) and (3) converge absolutely (see [1, 2]).
For , the -integral on is defined by
In particular, for , we set
In the theory of -analysis, the -analogue of a number is defined by
Now we review the definition of -Lebesgue spaces introduced by Fan and Zhao [13]. For and , we say a function belongs to if
The classical Morrey spaces, initially introduced by Morrey [33], play an important role in harmonic analysis and partial differential equations. Moreover, the central version of Morrey spaces are also well applied. We refer the readers to [34–39] for the studies of Morrey-type spaces. Here we define the -analogue of central Morrey-type spaces, which are called central -Morrey spaces.
Definition 1. Let , , and . A function belongs to the central -Morrey space if and only if where the supremum is taken over for all integers .
Obviously, if , then just reduces to the -Lebesgue space . We also point out that the central -Morrey spaces are closely related with the discrete Morrey spaces studied in [40, 41].
Now we are in a position to give the definition of -analogue of Hausdorff operators. For and , the -analogue of Hausdorff operator is defined by where .
By choosing and in the definition of , we recover the -analogue of Hardy operator and its dual operator , respectively:
Furthermore, if we take , then we obtain the -analogue of Hardy-Littlewood-Pólya operator:
Moreover, we arrive at the -analogue of Hilbert operator by taking in the definition of .
In the following section, we will establish the boundedness for the -analogue of Hausdorff operators on central -Morrey spaces. Moreover, we will show that the obtained constant is best possible. As corollaries, the sharp constants for -analogue of aforementioned operators on central -Morrey spaces are obtained in Section 3.
2. Main Results
The main result of this paper is the sharp estimates for the -analogue of Hausdorff operators on central -Morrey spaces .
Theorem 2. Let , , and . Assume that is a nonnegative function and . Then the following inequality holds, provided that Moreover, the constant in (13) is the best possible.
Proof. For any , using the definition given in (3), we can represent as
where .
For , by changing the variable , we have
Now assume holds (14). Using (16) and Minkowski’s inequality, we get
By taking for some integer in the above inequality, one has
As a consequence, for , by changing the variable , there holds
By taking the supremum over all integers , we arrive at
To show the constant is sharp, we take
We claim that . In fact, given for some integer , we get
Consequently, for any integer which yields
On the other hand
Therefore, for any integer , by changing variables and then , we have
The above estimates together with (24) show that
for any integer , which yields
by taking the supremum over all integers .
Equality (28) assures that the constant is sharp in (13).
Remark 3. We point out that for , Fan and Zhao ([13], Theorem 1.1) established the sharp constant for on , where and . This result, together with Theorem 2, gives the complete estimates for the -analogue of Hausdorff operators on central -Morrey spaces.
Theorem 2 can be applied to some concrete operators. The details will appear in the following section.
3. Applications
This section will give several applications of Theorem 2 by taking different in the definition of . The following corollaries give the sharp constants for the -analogue of Hardy operator and its dual operator on central -Morrey spaces.
Corollary 4. Let , , and . Then there holds Moreover, the constant in (29) is the best possible.
Proof. By taking in Theorem 2, it yields that is bounded on with the constant if is finite.
By a direct calculation, we get
The proof is finished.
Remark 5. Letting , Corollary 4 recovers the result of ([8], Theorem 2.1) formally, which gives the sharp constant for the -analogue of the Hardy operator on -Lebesgue spaces.
Corollary 6. Let , , and . Then there holds Moreover, the constant in (31) is the best possible.
Proof. By taking in Theorem 2, it yields that is bounded on with the constant if is finite.
A simple calculation yields
The proof is finished.
Remark 7. Letting , Corollary 4 recovers the result of ([12], Theorem 2) formally, which gave the sharp constant for the -analogue of the dual Hardy operator on -Lebesgue spaces.
Combining Corollary 4 with Corollary 6, we can deduce the sharp estimate for the -analogue of the Hardy-Littlewood-Pólya operator on central -Morrey spaces.
Corollary 8. Let , , and . Then there holds Moreover, the constant in (33) is the best possible.
Proof. By taking in Theorem 2, it yields that is bounded on with the constant if is finite. In view of the calculations in Corollary 4 with Corollary 6, we know that is finite and equal to The proof is finished.
Theorem 2 also yields the following estimate for the -analogue of the Hilbert operator on central -Morrey spaces.
Corollary 9. Let , , and . Then there holds Moreover, the constant in (35) is the best possible.
Proof. By taking in Theorem 2, it yields that is bounded on with the constant if is finite. By a direct computation, it yelds
which converges since .
The proof is finished.
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to express their deep gratitude to the anonymous referees for their careful reading of the manuscript and their comments and suggestions. This work was supported by the Nanhu Scholar Program for Young Scholars of Xinyang Normal University, the Natural Science Foundation of Henan Province (Grant No. 202300410338), the Shandong Jianzhu University Foundation (Grant No. X20075Z0101), and the Shandong Province Soft Science Research Project (Grant No. 2020RKB01671).