DOI: 10.1515/ans-2023-0218 ISSN: 1536-1365
Uniform volume estimates and maximal functions on generalized Heisenberg-type groups
Cheng Bi, Hong-Quan Li 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.