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

Authors: Mingquan Wei; Qianjun He; Xiang Li; Zequn Wang
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we establish the sharp constant forq-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.


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## Body

## 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 theq-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 Φt defined on 0,+∞, the Hausdorff operator HΦ is defined by
(1)HΦfx=∫0+∞Φx/yyfydy,x∈0,+∞.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 ofq-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)∫0xftdqt=1−qx∑k=0∞qkfqkxforx∈0,b,and the improper q-integral of a function f:0,+∞⟶ℝ is defined by the series
(3)∫0+∞ftdqt≔1−q∑k=−∞∞qkfqk,provided that the series on the right-hand sides of (2) and (3) converge absolutely (see [1, 2]).For0<a<b≤+∞, the q-integral on a,b is defined by
(4)∫abftdqt=∫0bftdqt−∫0aftdqt.In particular, forx∈0,+∞, we set
(5)∫x+∞ftdqt=∫0+∞ftdqt−∫0xftdqt.In the theory ofq-analysis, the q-analogue αq of a number α∈ℝ is defined by
(6)αq=1−qα1−q.Now we review the definition ofq-Lebesgue spaces introduced by Fan and Zhao [13]. For 0<q<1 and p∈0,+∞, we say a function f belongs to Lpdqt if
(7)fLpdqt≔∫0∞ftpdqt1/p<∞.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.
Let0<q<1, 1≤p<∞, and −1/p≤λ<0. A function f:0,+∞⟶ℝ belongs to the central q-Morrey space Ḃp,λdqt if and only if
(8)fḂp,λdqt≔supb−λ+1p·fχ0,bLpdqt<∞,where the supremum is taken over b=qm for all integers m.Obviously, ifλ=−1/p, then Ḃp,λdqt just reduces to the q-Lebesgue space Lpdqt. 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 ofq-analogue of Hausdorff operators. For 0<q<1 and Φ:0,+∞⟶ℝ, the q-analogue of Hausdorff operator hΦ is defined by
(9)hΦfx=∫0+∞Φx/ttftdqt,x∈0,+∞,where f:0,+∞⟶ℝ.By choosingΦt=1/tχ1,+∞t and Φt=χ0,1t in the definition of hΦ, we recover the q-analogue of Hardy operator H and its dual operator H∗, respectively:
(10)Hfx=1x∫0xftdqt,H∗fx=∫x∞fttdqt.Furthermore, if we takeΦt=1/tχ1,+∞t+χ0,1t, then we obtain the q-analogue of Hardy-Littlewood-Pólya operator:
(11)Pfx=∫0+∞ftmaxx,tdqt=Hfx+H∗fx.Moreover, we arrive at theq-analogue of Hilbert operator
(12)Tfx=∫0∞1x+tdqt,by taking Φt=1/1+tχ1,+∞t in the definition of hΦ.In the following section, we will establish the boundedness for theq-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,andT on central q-Morrey spaces are obtained in Section 3.


## 2. Main Results

The main result of this paper is the sharp estimates for theq-analogue of Hausdorff operators on central q-Morrey spaces Ḃp,λdqt.Theorem 2.
Let0<q<1, 1<p<+∞, and −1/p<λ<0. Assume that Φ is a nonnegative function and f∈Ḃp,λdqt. Then the following inequality
(13)hΦfḂp,λdqt≤C0fḂp,λdqt holds, provided that
(14)C0=∫0+∞Φttλ+1dqt<+∞.
Moreover, the constantC0 in (13) is the best possible.Proof.
For anyf∈Ḃp,λdqt, using the definition given in (3), we can represent hΦf as
(15)hΦfx=1−q∑j=−∞+∞Φxqjfqj,where x∈0,+∞.
For0<b<+∞, by changing the variable k=l−j, we have
(16)hΦfχ0,bLpdqt=∫0bhΦfxpdqx1/p=1−q∫0b∑j=−∞+∞Φxqjfqjpdqx1/p=1−q1+1/pb∑l=0+∞ql∑j=−∞+∞Φqlqjbfqjp1/p=1−q1+1/pb1/p∑l=0+∞ql∑k=−∞+∞Φqkbfql−kp1/p.
Now assume holds (14). Using (16) and Minkowski’s inequality, we get
(17)hΦfχ0,bLpdqt≤1−q1+1/pb1/p∑k=−∞+∞Φqkb∑l=0+∞qlfql−kp1/p.
By takingb=qm for some integer m in the above inequality, one has
(18)hΦfχ0,qmLpdqt≤1−q1+1/pqm/p∑k=−∞+∞Φqk+m∑l=0+∞qlfqlq−kp1p=1−qqm/p∑k=−∞+∞Φqk+mqk/p1−qq−k∑l=0+∞qlfqlq−kp1p=1−qqm/p∑k=−∞+∞Φqk+mqk/pfχ0,q−kLpdqt.
As a consequence, forb=qm, by changing the variable i=k+m, there holds
(19)b−λ+1/phΦfχ0,bLpdqt≤1−q∑k=−∞+∞Φqk+mqk/pqmλfχ0,q−kLpdqt=1−q∑k=−∞+∞Φqk+mqk+mλqkλ+1/pfχ0,q−kLpdqt≤1−q∑i=−∞+∞qiΦqiqiλ+1·fḂp,λdqt=∫0∞Φttλ+1dqt·fḂp,λdqt.
By taking the supremum over all integersm, we arrive at
(20)hΦfḂp,λdqt≤∫0∞Φttλ+1dqt·fḂp,λdqt.
To show the constantC0=∫0∞Φt/tλ+1dqt is sharp, we take
(21)f0x=xλ,x∈0,+∞.
We claim thatf0∈Ḃp,λdqt. In fact, given b=qm for some integer m, we get
(22)f0χ0,qmLpdqtp=∫0qmxpλdqt=1−qqm∑k=0+∞qkxk+mpλ=1−q∑l=m+∞qlxlpλ=1−q∑l=m+∞q1+pλl=1−qq1+pλm1−q1+pλ.
Consequently, for any integerm(23)q−mλ+1/pf0χ0,qmLpdqt=1−q1−q1+pλ1/p,which yields
(24)f0Ḃp,λdqt=1−q1−q1+pλ1/p<+∞.
On the other hand(25)hΦf0x=1−q∑i=−∞+∞Φxqiqiλ.
Therefore, for any integerm, by changing variables i=j+m−l and then k=m+j, we have
(26)∫0qmhΦf0xpdqx=1−qp+1qm∑j=0∞qj∑i=−∞+∞Φqj+m−iqiλp=1−q∑j=0∞qm+j1−q∑i=−∞+∞Φqj+m−iqiλp=1−q∑j=0∞qm+j1−q∑l=−∞+∞Φqlqj+m−lλp=1−q∑j=0∞qm+j1+pλ1−q∑l=−∞+∞qlΦqlqlλ+1p=1−q∑k=m∞qk1+pλ∫0∞Φttλ+1dqtp=1−qqm1+pλ1−q1+pλ∫0∞Φttλ+1dqtp.
The above estimates together with (24) show that
(27)q−m1+pλ∫0qmhΦf0xpdqx=1−q1−q1+pλ∫0∞Φttλ+1dqtp=∫0∞Φttλ+1dqtp·f0Ḃp,λdqtp,for any integer m, which yields
(28)hΦf0Ḃp,λdqt=∫0∞Φttλ+1dqt·f0Ḃp,λdqt,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 forλ=−1/p, Fan and Zhao ([13], Theorem 1.1) established the sharp constant for hΦ on Lqdqt, 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.Theorem2 can be applied to some concrete operators. The details will appear in the following section.


## 3. Applications

This section will give several applications of Theorem2 by taking different Φ in the definition of hΦ. 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.
Let0<q<1, 1<p<+∞, and −1/p<λ<0. Then there holds
(29)HfḂp,λdqt≤11+λqfḂp,λdqt.
Moreover, the constant1/1+λq in (29) is the best possible.Proof.
By takingΦt=1/tχ1,∞t in Theorem 2, it yields that H is bounded on Ḃp,λdqt with the constant C1=∫0+∞χ1,+∞t/t2+λdqt if C1 is finite.
By a direct calculation, we get(30)C1=∫0+∞χ1,+∞tt2+λdqt=1−q∑k=−∞+∞qkq−k2+λχ1,+∞qk=1−q∑k=−∞0qkq−k2+λ=1−q1−qλ+1=11+λq<+∞.
The proof is finished.Remark 5.
Lettingλ⟶−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.
Let0<q<1, 1<p<+∞, and −1/p<λ<0. Then there holds
(31)H∗fḂp,λdqt≤1−λqfḂp,λdqt.
Moreover, the constant1/−λq in (31) is the best possible.Proof.
By takingΦt=χ0,1t in Theorem 2, it yields that H∗ is bounded on Ḃp,λdqt with the constant C2=∫011/t1+λdqt if C2 is finite.
A simple calculation yields(32)C2=∫011t1+λdqt=1−q∑k=0+∞qkq−k1+λ=1−q∑k=0+∞q−kλ=1−q1−q−λ=1−λq<+∞.
The proof is finished.Remark 7.
Lettingλ⟶−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 Corollary4 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.
Let0<q<1, 1<p<+∞, and −1/p<λ<0. Then there holds
(33)PfḂp,λdqt≤11+λq+1−λqfḂp,λdqt.
Moreover, the constant1/1+λq+1/−λq in (33) is the best possible.Proof.
By takingΦt=1/tχ1,+∞t+χ0,1t in Theorem 2, it yields that P is bounded on Ḃp,λdqt with the constant C3=C1+C2=∫1+∞1/t2+λdqt+∫011/t1+λdqt 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)11+λq+1−λq.
The proof is finished.Theorem2 also yields the following estimate for the q-analogue of the Hilbert operator on central q-Morrey spaces.Corollary 9.
Let0<q<1, 1<p<+∞, and −1/p<λ<0. Then there holds
(35)TfḂp,λdqt≤1−q∑k=−∞01qkλ1+qkfḂp,λdqt.
Moreover, the constant1−q∑k=−∞01/qkλ1+qk in (35) is the best possible.Proof.
By takingΦt=1/1+tχ1,+∞t in Theorem 2, it yields that P is bounded on Ḃp,λdqt with the constant C4=∫0+∞χ1,+∞t/t1+λ1+tdqt if C4 is finite. By a direct computation, it yelds
(36)C4=∫0+∞χ1,+∞tt1+λ1+tdqt=1−q∑k=−∞+∞qkχ1,+∞qkqk1+λ1+qk=1−q∑k=−∞01qkλ1+qk,which converges since λ<0.
The proof is finished.



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*Source: 4879613-2022-05-23.xml*