On the Kor\'anyi Spherical maximal function on Heisenberg groups

TL;DR

This paper establishes sharp $L^p$ to $L^q$ bounds for maximal and spherical mean operators related to the Korányi sphere in Heisenberg groups, advancing understanding of their boundedness and sparse domination properties.

Contribution

It provides the first sharp $L^p$ to $L^q$ estimates for these operators, improving bounds on sparse domination in the context of Heisenberg groups.

Findings

Sharp $L^p$ to $L^q$ estimates for local maximal operators

Enhanced sparse domination bounds for global maximal operators

Improved bounds for spherical means over the Korányi sphere

Abstract

We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the K\'oranyi 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 $L^p\to L^q$ estimates for spherical means over the Kor\'anyi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.