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

Abstract

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 ${\mathbb{R}}^{2n}$. Consider the maximal function defined by $Mf={\sup }_{t>0}|f\ast {\mu }_t|$. We prove for $n\ge 2$ that $M$ defines an operator bounded on $L^p(H^n)$ provided that $p>2n/(2n-1)$. 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.

arXiv commentESI preprint 1292, March 2003