Centered Hardy–Littlewood maximal functions on H-type groups revisited

In this paper, using the method of stationary phase, we obtain the uniform asymptotic behavior of the Poisson kernel, associated to the canonical sub-Laplacian as well as the full Laplacian, on Heisenberg-type groups \( \mathbb {H} ( 2n, m )\) $\mathbb{H}(2n,m)$ . We prove that there exists a constant \(C > 0\) $C>0$ , independent of (n, m), such that \(\Vert M_K \Vert _{L^1 \rightarrow L^{1, \infty }} \le C \, n\) $\Vert M_K{\Vert }_{L^1\to L^{1,\infty }}\le Cn$ , where \(M_K\) $M_K$ denotes the centered Hardy-Littlewood maximal operator defined by the Korányi norm. While for \(M = M_{CC}\) $M=M_{\mathit{CC}}$ or \(M_R\) $M_R$ , the corresponding operator related to the canonical sub-Riemannian and Riemannian distance respectively, we obtain \(\Vert M \Vert _{L^1 \rightarrow L^{1, \infty }} \le C \, ( 3 / 2 )^{ \frac{ m }{ 2 }} \, n\) ${\Vert M\Vert }_{L^1\to L^{1,\infty }}\le C{(3/2)}^{\frac{m}{2}}n$ . In particular, we provide an affirmative answer to the question left open in Li and Qian (Trans Am Math Soc 366:1497–1524, 2014) [22] by means of a much simpler method. Besides, these bounds are perfectly matched with the associated Green function. Furthermore, the \(( 3 / 2 )^{ \frac{ m }{ 2 }} \, n\) ${(3/2)}^{\frac{m}{2}}n$ order bound remains uniformly valid, whenever the canonical Sub-Riemannian or Riemannian distance are replaced by a large class of Carnot-Carathéodory distances.