On the Korányi Spherical maximal function on Heisenberg groups

Abstract

We prove L p → L q estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp L p → L q estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in [10] for the associated lacunary maximal operator.

Spherical maximal operators on Heisenberg groups: Restricted dilation sets

“…Finally we remark that the behavior of maximal operators associated with the codimension two spherical means considered here is quite different from the behavior of maximal functions associated with hypersurfaces in the Heisenberg group. Of particular interest here is the Korányi sphere, for which the sharp L p improving properties of the local full maximal operator up to endpoints were obtained in a recent paper by one of the authors [27], see also partial results about averages in previous work [11] by Ganguly and Thangavelu.…”