Weighted Morrey estimates for Hausdorff operator and its commutator on the Heisenberg group

Abstract

In this paper, we study the high-dimensional Hausdorff operators, defined via a general linear mapping A, and their commutators on the weighted Morrey spaces in the setting of the Heisenberg group. Particularly, under some assumption on the mapping A, we establish their sharp boundedness on the power weighted Morrey spaces.

Weighted CBMO estimates for the commutators of matrix Hausdorff operator on Heisenberg group

“…Recently, weighted boundedness of matrix Hausdorff operators and their commutators defined on different underlying spaces are established in [34,35,36,37,38,39,40,41].…”

Boundedness of Hausdorff operators on real Hardy spaces H1 over locally compact groups

“…2) Special cases of Hausdorff operator on H n were considered in [27]. (4) Strict upper triangular groups [12, p. 6, 7], [2, Subsection VII.3.3].…”

“…The survey article by E. Liflyand [19] contains main results on Hausdorff operators in various settings and bibliography up to 2013. See also [5], and [24], [6], [7], [27], [30]. The recent paper by Ruan and Fan [26] contains in particular several sharp conditions for boundedness of Hausdorff operators on the space H 1 (R n ).…”

The WeightedLpandBMOEstimates for Fractional Hausdorff Operators on the Heisenberg Group

“…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][17][18][19][20][21][22][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.…”