Spherical maximal operators on Heisenberg groups: Restricted dilation sets

TL;DR

This paper investigates the boundedness of spherical maximal operators on Heisenberg groups with restricted dilation sets, establishing $L^p$ to $L^q$ estimates that depend on the dimension of the dilation set.

Contribution

It introduces new $L^p$ to $L^q$ bounds for spherical maximal functions on Heisenberg groups with restricted dilations, linking estimates to the dimension of the dilation set.

Findings

Established $L^p$ to $L^q$ estimates for restricted dilation maximal functions

Connected the estimates to various notions of dimension of the dilation set

Extended understanding of spherical averages on Heisenberg groups

Abstract

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions $M_Ef$ where the dilations are restricted to a set $E$. We prove $L^p\to L^q$ estimates for these maximal operators; the results depend on various notions of dimension of $E$.