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

Abstract

Let $G\cong\mathbb{R}^{d} \ltimes \mathbb{R}$ be a finite-dimensional two-step nilpotent group with the group multiplication $(x,u)\cdot(y,v)\rightarrow(x+y,u+v+x^{T}Jy)$ where $J$ is a skew-symmetric matrix satisfying a degeneracy condition with $2\leq {\rm rank}\, J <d$. Consider the maximal function defined by $$ {\frak M}f(x, u)=\sup_{t>0}\big|\int_{\Sigma} f(x-ty, u- t x^{T}Jy) d\mu(y)\big|, $$ where $\Sigma$ is a smooth convex hypersurface and $d\mu$ is a compactly supported smooth density on $\Sigma$ such that the Gaussian curvature of $\Sigma$ is nonvanishing on supp $d\mu$. In this paper we prove that when $d\geq 4$, the maximal operator ${\frak M}$ is bounded on $L^{p}(G)$ for the range $(d-1)/(d-2)<p\leq\infty$.