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Theoretical study demonstrates uniform volume bounds on generalized Heisenberg-type groups, indicating extended weak (1, 1) maximal function estimates.
On generalized Heisenberg-type groups G ( 2 n , m , U , W ) G(2n,m,U,W) , 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.
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Bi et al. (2026) studied this question.
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