Spherical means on the Heisenberg group: Stability of a maximal function estimate

TL;DR

This paper proves the $L^p$-boundedness of a maximal function associated with spherical measures on the Heisenberg group, using advanced decoupling inequalities to establish optimal bounds.

Contribution

It introduces a novel application of decoupling inequalities to the Heisenberg group setting for maximal functions, achieving optimal $L^p$ bounds.

Findings

Proves $L^p$ boundedness of the maximal function in an optimal range.

Utilizes decoupling inequalities by Wolff and Bourgain-Demeter.

Establishes stability estimates for spherical means on the Heisenberg group.

Abstract

Consider the surface measure $\mu$ on a sphere in a nonvertical hyperplane on the Heisenberg group $\mathbb{H}^n$, $n\ge 2$, and the convolution $f*\mu$. Form the associated maximal function $Mf=\sup_{t>0}|f*\mu_t|$ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain-Demeter to prove $L^p$-boundedness of $M$ in an optimal range.