Singular spherical maximal operators on a class of two step nilpotent Lie groups

Let $H^n\cong {\mathbb{R}}^{2n}⋉\mathbb{R}$ be the Heisenberg group and let ${\mu }_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$\mathrm{.}Considerthemaximalfunctiondefinedby$Mf=\sup_{t>0} |f*\mu_t|$. We prove for $n\ge 2$ that $M$ defines an operator bounded on $L^p(H^n)$ provided that $p>2n\mathrm{/}(2n-1)$. This improves an earlier result by Nevo and Thangavelu, and the range for $L^p$ boundedness is optimal. We also extend the result to a more general setting of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.