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

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-Carathéodory 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.

22 pages, incporporating suggestions from referees reports, accepted for publication in Adv. Nonlinear Stud., Special Issue in honor of Professor Leonard Gross's 95th birthday