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By getting uniformly asymptotic estimates for the Poisson kernel on Heisenberg groups ${ℍ}_{2n+1}$, we prove that there exists a constant A > 0, independent of n ∈ ℕ*, such that for all $f\in L¹\left({ℍ}_{2n+1}\right)$, we have ${\left|\right|Mf\left|\right|}_{L^{1,\infty }}\le An\left|\right|f\left|\right|₁$, where M denotes the centered Hardy-Littlewood maximal function defined by the Carnot-Carathéodory distance or by the Korányi norm.