Analysis & Applications Seminar

Centered Hardy-Littlewood Maximal Functions on H-Type Groups Revisited

Speaker:  Hongquan Li (Fudan University)
Date:  Tuesday, 20 June 2023 - 11:30
Place:  Aula Gris 2, ICMAT


Using the method of stationary phase, we obtain the uniform asymptotic behavior of the Poisson kernel, associated to the sub-Laplacian as well as the full Laplacian, on Heisenberg-type groups \(\mathbb{H} (2n, m)\). Then we prove that there exists a constant \(C>0\) independent of \((n, m)\) such that \(\|M_K\|_{L^1 \longrightarrow L^{1, \infty}} \le C \, n\),  where \(M_K\) denotes the centered Hardy-Littlewood maximal function defined by the Kor\'anyi norm. While for \(M = M_{CC}\) or \(M_R\), the centered  Hardy-Littlewood maximal function related to the canonical sub-Riemannian and Riemannian distance respectively, we obtain \(\| M \|_{L^1 \longrightarrow L^{1, \infty}} \le C  \, (3/2 )^{\frac{m}{2}} \, n\). These bounds are perfectly matched with the associated Green's function. This talk is based on a work with Chen Bi and Ye Zhang.