In this paper, we establish the sharp constant for q-analogus of Hausdorff operators on central q-Morrey spaces. As applications, the sharp constants for the q-analogus of Hardy operator and its dual operator, the q-analogue of Hardy-Littlewood-Pólya operator, and the q-analogue of Hilbert operator on central q-Morrey spaces are deduced.

1. Introduction

The quantum calculus (q-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 q-calculus setting, such as the Hardy q-inequalities [8–11], the Hausdorff q-inequalities [12, 13], and some other important q-integral inequalities [14, 15]. The aim of this paper is to study the sharp constant for Hausdorff operators in the framework of q-calculus on central q-Morrey spaces. We first recall the definition of Hausdorff operators on Euclidean spaces. For a given locally integrable function Φ(t) defined on (0, +∞), the Hausdorff operator H_Φ is defined by

(1)

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 q-inequalities and obtained the sharp constants, which extends the corresponding results of the classical Hardy inequalities. Subsequently, Guo and Zhao [12] introduced the q-analogue of Hausdorff operators and established the Hausdorff q-inequalities, which generalized the main results in [8]. One can also consult [13] for the Hausdorff q-inequalities for the multivariable setting. Note that the main results of [12, 13] essentially build the sharp constants for q-analogue of Hausdorff operators on q-Lebesgue spaces. It is natural to ask whether the q-analogue of Hausdorff operators can be bounded on central q-Morrey spaces (see the definition below).

To this end, we first introduce some basic notations and definitions of q-calculus, which are necessary for understanding this paper. Fix a positive number q ∈ (0, 1) and a function f : [0, b)⟶ℝ, 0 < b ≤ +∞, the q-integral of the q-Jackson integral of f is defined by

(2)

and the improper q-integral of a function f : [0, +∞)⟶ℝ is defined by the series

(3)

provided that the series on the right-hand sides of (2) and (3) converge absolutely (see [1, 2]).

For 0 < a < b ≤ +∞, the q-integral on [a, b] is defined by

(4)

In particular, for x ∈ (0, +∞), we set

(5)

In the theory of q-analysis, the q-analogue [α]_q of a number α ∈ ℝ is defined by

(6)

Now we review the definition of q-Lebesgue spaces introduced by Fan and Zhao [13]. For 0 < q < 1 and p ∈ (0, +∞), we say a function f belongs to L^p(d_qt) if

(7)

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 q-analogue of central Morrey-type spaces, which are called central q-Morrey spaces.

Definition 1. Let 0 < q < 1, 1 ≤ p < ∞, and −(1/p) ≤ λ < 0. A function f : [0, +∞)⟶ℝ belongs to the central q -Morrey space if and only if

(8)

where the supremum is taken over b = q^m for all integers m.

Obviously, if λ = −1/p, then just reduces to the q-Lebesgue space L^p(d_qt). We also point out that the central q-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 q-analogue of Hausdorff operators. For 0 < q < 1 and Φ : [0, +∞)⟶ℝ, the q-analogue of Hausdorff operator h_Φ is defined by

(9)

where f : [0, +∞)⟶ℝ.

By choosing Φ(t) = (1/t)χ_(1,+∞)(t) and Φ(t) = χ_{(0, 1)}(t) in the definition of h_Φ, we recover the q-analogue of Hardy operator H and its dual operator H^∗, respectively:

(10)

Furthermore, if we take Φ(t) = (1/t)χ_(1,+∞)(t) + χ_{(0, 1)}(t), then we obtain the q-analogue of Hardy-Littlewood-Pólya operator:

(11)

Moreover, we arrive at the q-analogue of Hilbert operator

(12)

by taking Φ(t) = (1/(1 + t))χ_[1,+∞)(t) in the definition of h_Φ.

In the following section, we will establish the boundedness for the q-analogue of Hausdorff operators on central q-Morrey spaces. Moreover, we will show that the obtained constant is best possible. As corollaries, the sharp constants for q-analogue of aforementioned operators H, H^∗, P, and T on central q-Morrey spaces are obtained in Section 3.

2. Main Results

The main result of this paper is the sharp estimates for the q-analogue of Hausdorff operators on central q-Morrey spaces .

Theorem 2. Let 0 < q < 1, 1 < p < +∞, and −(1/p) < λ < 0. Assume that Φ is a nonnegative function and . Then the following inequality

(13)

holds, provided that

(14)

Moreover, the constant C₀ in (13) is the best possible.

Proof. For any , using the definition given in (3), we can represent h_Φf as

(15)

where x ∈ (0, +∞).

For 0 < b < +∞, by changing the variable k = l − j, we have

(16)

Now assume holds (14). Using (16) and Minkowski’s inequality, we get

(17)

By taking b = q^m for some integer m in the above inequality, one has

(18)

As a consequence, for b = q^m, by changing the variable i = k + m, there holds

(19)

By taking the supremum over all integers m, we arrive at

(20)

To show the constant is sharp, we take

(21)

We claim that . In fact, given b = q^m for some integer m, we get

(22)

Consequently, for any integer m

(23)

which yields

(24)

On the other hand

(25)

Therefore, for any integer m, by changing variables i = j + m − l and then k = m + j, we have

(26)

The above estimates together with (24) show that

(27)

for any integer m, which yields

(28)

by taking the supremum over all integers m.

Equality (28) assures that the constant C₀ is sharp in (13).

Remark 3. We point out that for λ = −(1/p), Fan and Zhao ([13], Theorem 1.1) established the sharp constant for h_Φ on L^q(d_qt), where 0 < q < 1 and 1 < p < +∞. This result, together with Theorem 2, gives the complete estimates for the q-analogue of Hausdorff operators on central q-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 Φ in the definition of h_Φ. The following corollaries give the sharp constants for the q-analogue of Hardy operator and its dual operator on central q-Morrey spaces.

Corollary 4. Let 0 < q < 1, 1 < p < +∞, and −(1/p) < λ < 0. Then there holds

(29)

Moreover, the constant 1/([1 + λ]_q) in (29) is the best possible.

Proof. By taking Φ(t) = (1/t)χ_(1,∞)(t) in Theorem 2, it yields that H is bounded on with the constant if C₁ is finite.

By a direct calculation, we get

(30)

The proof is finished.

Remark 5. Letting λ⟶−(1/p), Corollary 4 recovers the result of ([8], Theorem 2.1) formally, which gives the sharp constant for the q -analogue of the Hardy operator on q -Lebesgue spaces.

Corollary 6. Let 0 < q < 1, 1 < p < +∞, and −(1/p) < λ < 0. Then there holds

(31)

Moreover, the constant 1/[−λ]_q in (31) is the best possible.

Proof. By taking Φ(t) = χ_{(0, 1)}(t) in Theorem 2, it yields that H^∗ is bounded on with the constant if C₂ is finite.

A simple calculation yields

(32)

The proof is finished.

Remark 7. Letting λ⟶−(1/p), Corollary 4 recovers the result of ([12], Theorem 2) formally, which gave the sharp constant for the q -analogue of the dual Hardy operator on q -Lebesgue spaces.

Combining Corollary 4 with Corollary 6, we can deduce the sharp estimate for the q-analogue of the Hardy-Littlewood-Pólya operator on central q-Morrey spaces.

Corollary 8. Let 0 < q < 1, 1 < p < +∞, and −(1/p) < λ < 0. Then there holds

(33)

Moreover, the constant 1/[1 + λ]_q + 1/[−λ]_q in (33) is the best possible.

Proof. By taking Φ(t) = (1/t)χ_(1,+∞)(t) + χ_{(0, 1)}(t) in Theorem 2, it yields that P is bounded on with the constant if C₃ is finite. In view of the calculations in Corollary 4 with Corollary 6, we know that C₃ is finite and equal to

(34)

The proof is finished.

Theorem 2 also yields the following estimate for the q-analogue of the Hilbert operator on central q-Morrey spaces.

Corollary 9. Let 0 < q < 1, 1 < p < +∞, and −(1/p) < λ < 0. Then there holds

(35)

Moreover, the constant in (35) is the best possible.

Proof. By taking Φ(t) = (1/1 + t)χ_[1,+∞)(t) in Theorem 2, it yields that P is bounded on with the constant if C₄ is finite. By a direct computation, it yelds

(36)

which converges since λ < 0.

The proof is finished.

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).

Open Research

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Sharp Constants for q-Analogue of Hausdorff Operators on Central q-Morrey Spaces

Abstract

1. Introduction

2. Main Results

3. Applications

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

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Sharp Constants for q-Analogue of Hausdorff Operators on Central q-Morrey Spaces

Abstract

1. Introduction

2. Main Results

3. Applications

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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