Uniform volume estimates and maximal functions on generalized Heisenberg-type groups

TL;DR

This paper provides uniform volume estimates and maximal function bounds on generalized Heisenberg-type groups, extending previous results and establishing volume doubling properties for certain metrics.

Contribution

It introduces new uniform volume estimates and maximal function bounds on generalized Heisenberg groups, extending prior work and including volume doubling results.

Findings

Established uniform volume estimates for Carnot-Carathéodory balls.

Proved weak (1,1) maximal inequalities with explicit bounds.

Demonstrated volume doubling properties for specific Riemannian metrics.

Abstract

On generalized Heisenberg-type groups $\mathbb{G}(2n,m,\mathbb{U},\mathbb{W})$, we give uniform volume estimates for the ball defined by a large class of Carnot-Carath\'{e}odory distances, and establish weak (1, 1) $O(C^m \, n)$-estimates for associated centered Hardy-Littlewood maximal functions, extending the results in \cite{BLZ25}. As a by-product, we establish uniformly volume doubling property on Heisenberg groups for a class of left-invariant Riemannian metrics.