An optimal theorem for the spherical maximal operator on the Heisenberg group

Let\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{I}^n = \mathbb{C}^n \times \mathbb{R}$$ \end{document} be the Heisenberg group and μr be the normalized surface measure on the sphere of radiusr in ℂn. Let\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Mf = \sup _{r > 0} \left| {f * \mu _r } \right|$$ \end{document}. We prove an optimalLp-boundedness result for the spherical maximal functionMf, namely we prove thatM is bounded onLp(In) if and only ifp>2n/2n−1.