Abstract
In the setting of Heisenberg group, we characterize those functions , for which the fractional Hausdorff operators and Hausdorff operators , are bounded on spaces with power weights, space, and Hardy spaces, respectively. Meanwhile, the corresponding operator norms of and are worked out.
1. Introduction
The Hausdorff operator has a deep root in the study of the Fourier analysis. Particularly, one-dimensional Hausdorff operator in Euclidean space is closely related to the summability of the classical Fourier series. Modern theory of Hausdorff operator started with the work of Siskakis in the complex analysis setting and with the work of Liflyand–Móricz in the Fourier transform setting (for more information about the background and the development of the Hausdorff operator one can refer to [1–5] and references therein).
Fix an appropriate function on , the one-dimensional Hausdorff operator [5, 6] is defined formally by
For , by a change of variables, one has
Naturally, the n-dimensional Hausdorff operators on are defined by
It is well known that many classical operators in harmonic analysis are special cases of Hausdorff operators if one chooses suitable (see, for example, [1, 6–10]), including Hardy operators and Cesàro operator [11–14]. The Hardy–Littlewood–Pólya operator and the Riemann–Liouville fractional integral can also be derived from the Hausdorff operators. Besides, if we set in and in , then and become the weighted Hardy–Littlewood average defined by Xiao in [15], which is given by
And, if we set and in and , respectively, then they become the weighted Cesàro average in [15], which is defined by
In recent years, there are many works about the Hardy operators, weighted Hardy operators, and Hausdorff operators in the setting of the Heisenberg group (see, for example, [16–23]). It is known that the Heisenberg group plays an important role in several branches of mathematics such as harmonic analysis, representation theory, several complex analysis, partial differential equation, and quantum mechanics. In this paper, we will focus on the boundedness of Hausdorff operators on power-weighted Lebesgue spaces in the setting of the Heisenberg group , such group can be identified with with the group structure (for all the notation on , we refer to Section 2).
Let be a locally integrable function on . The Hausdorff operators on are defined by
The fractional Hausdorff operator on is defined by
Remark 1. It is clear that when , then . And, if we choose and in the operator , respectively, we can obtain the Hardy operators and their adjoint on the Heisenberg group [22]:We obtain the following weak and strong boundedness for fractional Hausdorff operators on Lebesgue spaces with power weights (for Euclidean version, one can refer to Gao et al. [9]).
Theorem 1. Let , , , and . LetIf is a radial function and for , then
Theorem 2. Let , , and satisfy . Ifwhere s satisfies , then
Corollary 1. Let , , and satisfy . For any radial function , ifwhere s satisfies , then
Corollary 2. Suppose that and and . Ifwhere , thenFor Hausdorff operators, we get the best estimates for them on Lebesgue spaces with power weights. Chuong et al. [17] also gave the approximation of the norm of these operators on such spaces.
Theorem 3. Let , , and . Then, is a bounded operator on if and only ifMoreover, when (17) holds, then
Remark 2. From Theorem 3, we can see that if is radial, thenAnd, Theorem 3 reduces to Theorem 2 in [18].
Theorem 4. Let , and . Then(i) is bounded on if and only if Moreover, when (17) holds, then(ii) is bounded on if and only ifMoreover, when (22) holds, thenSince the dual space of is isomorphic to (c.f. [24]), we can actually deduce the following results.
Corollary 3. Let . Then,
is bounded on if and only ifMoreover, when (24) holds, thenSuppose is a function. For a measurable function f on , we define the weighted Hardy operator asand define the weighted Cesàro operator asIt is easy to check that when , then . When , then . Therefore, we can get the sharp conditions for boundedness of weighted Hardy operators, and their Euclidean case is due to Xiao [15].
Corollary 4. Let be a function and let , . Then,(i) is bounded on if and only if Moreover, when (28) holds, the operator norm of on is given by(ii) is bounded on if and only ifMoreover, when (30) holds, the operator norm of on is given by
Corollary 5. Let be a function. Then,(i) is bounded on if and only if Moreover, when (32) holds, the operator norm of on is given by(ii) is bounded on if and only ifMoreover, when (34) holds, the operator norm of on is given byAlso, by the duality of and , we can obtain the sufficient and necessary conditions for boundedness of weighted Hardy and Cesàro operators on Hardy space .
Corollary 6. Let be a function. Then,(i) is bounded on if and only if Moreover, when (36) holds, the operator norm of on is given by(ii) is bounded on if and only ifMoreover, when (38) holds, the operator norm of on is given byThe paper is organized as follows. In Section 2, we recall the notation and definitions related to the Heisenberg group. In Section 3, we will discuss the boundedness for fractional Hausdorff operators on Lebesgue spaces with power weights, and give the proof of Theorem 1–4.
2. Preliminaries and Notations
Recall that the Heisenberg group is a noncommutative nilpotent Lie group, with the underlying manifold , whose group law can be written as
By definition, we can see that the identity element on is , while the reverse element of x is . The corresponding Lie algebra is generated by the left-invariant vector fields:
The only nontrivial commutator relations are
is a homogeneous group with dilations
The Haar measure on coincides with the usual Lebesgue measure on . We denote the measure of any measurable set by . Then,where is called the homogeneous dimension of .
The Heisenberg distance derived from the normis given by
This distance d is left-invariant in the sense that remains unchanged when p and q are both left-translated by some fixed vector on . Furthermore, d satisfies the triangular inequality (see P. 320 in [25])
For and , the ball and sphere with center x and radius r on are given byrespectively. And, we havewhereis the volume of the unit ball on . The area of on is [26] (for more details about Heisenberg group, see [24, 25]).
Let be the space of all measurable functions f on such that
The bounded mean oscillation space is defined to be the space of all locally integrable functions f on such thatwhere the supremum is taken over all balls in and .
If f and are measurable functions on , their convolution is defined by
Let denote the Schwartz class on the Heisenberg group . A measurable complex-valued function f on belongs to the Hardy space if and only ifwithwhere is some fixed function in with and denotes the dilation .
3. Boundedness for Fractional Hausdorff Operators on the Lebesgue Spaces with Power Weights
In this section, we will give the proof of Theorem 1–4.
Proof of Theorem 1. Since is radial, we can writeIf , by Hölder’s inequality, we havewhere and .
If , we haveLet and , for any and , we haveTherefore,Consequently,This completes the proof.
Proof of Theorem 2. By changing to polar coordinates and the Fubini theorem, we haveBy Minkowski’s inequality, we obtainHere,can be regarded as a convolution in the multiplicative group with Haar measure . By Young’s inequality, we haveTherefore,Theorem 2 is proved.
Proof of Theorem 3. The sufficient part can be immediately deduced by choosing and in Theorem 2.
Next, we will give the proof of necessary part. If is a bounded operator on , then there exists a constant such thatFor any , takeIt is easy to check that andBy changing variables, we haveThus,Therefore,Letting on both sides, we getWhen (17) holds, from Theorem 2, we can see thatOn the contrary, by (73), we havewhich implies (18). Theorem 3 is proved.
Proof of Theorem 4. (i)For any , by the Minkowski inequality, we have Therefore, if (20) holds, then is bounded on and On the contrary, suppose is bounded on , and denote . Like in Theorem 1.3, for any , we also take Then, Therefore, The rest of the proof is the same as that of Theorem 3.(ii)For each and ball , let be the ball , then .Suppose (22) holds. Let , and let B be a ball. Then, by the Fubini theorem, we haveThen, using the Fubini theorem again, we getwhich implies that is bounded on .
Conversely, if is bounded on . ChooseThen, with . And,Consequently,Moreover, when (22) holds, then (83) and (85) imply thatThis completes the proof.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was partially supported by the Applied Research Project of Qingdao Postdoctoral (2019-47), the National Natural Science Foundation of China (grant nos. 11671185 and 11701250), the Natural Science Foundation of Shandong Province (grant nos. ZR2018LA002 and ZR2019YQ04).