Weighted Herz space estimates for Hausdorff operators on the Heisenberg group

Abstract

In this article, we study the Hausdorff operator, defined via a general linear mapping $A$, on weighted Herz spaces in the setting of the Heisenberg group. Under some assumptions on the mapping $A$, we establish its sharp boundedness on power-weighted Herz spaces and power-weighted Lebesgue spaces in the Heisenberg group. Our proof is heavily based on the block decomposition of the Herz space, which is quite different from any other function spaces. Our results extend and improve some existing theorems.