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

Abstract On generalized Heisenberg-type groups G ( 2 n , m , U , W ) $\mathbb{G}\left(2n,m,\mathbb{U},\mathbb{W}\right)$ , we establish uniform volume estimates for the ball associated with a large class of Carnot-Carathéodory distances. As an application, we prove weak (1, 1) O ( C m n )-estimates for the corresponding centered Hardy-Littlewood maximal functions, extending the results in [C. Bi, H.-Q. Li, and Y. Zhang, “Centered Hardy-Littlewood maximal functions on H-type groups revisited,” Math. Ann. , vol. 391, no. 3, pp. 3765–3797, 2025]. As a by-product, we obtain uniformly volume doubling property on Heisenberg groups for a natural family of left-invariant Riemannian metrics.