Spherical maximal functions on two step nilpotent Lie groups

TL;DR

This paper extends the boundedness results of spherical maximal functions to all two-step nilpotent Lie groups with dimension d≥3, removing previous nondegeneracy restrictions.

Contribution

It proves sharp L^p bounds for spherical maximal functions on all two-step nilpotent Lie groups with d≥3, generalizing prior results on Métivier groups.

Findings

Established sharp L^p bounds for all two-step nilpotent Lie groups with d≥3.

Removed the nondegeneracy condition previously required in known results.

Extended the scope of spherical maximal function analysis beyond Métivier groups.

Abstract

Consider $\mathbb R^d\times \mathbb R^m$ with the group structure of a two-step nilpotent Lie group and natural parabolic dilations. The maximal function originally introduced by Nevo and Thangavelu in the setting of the Heisenberg group deals with noncommutative convolutions associated to measures on spheres or generalized spheres in $\mathbb R^d$. We drop the nondegeneracy condition in the known results on M\'etivier groups and prove the sharp $L^p$ boundedness result for all two step nilpotent Lie groups with $d\ge 3$.