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

Abstract

Let $G\cong {\mathbb{R}}^d⋉\mathbb{R}$ be a finite-dimensional two-step nilpotent group with the group multiplication $(x,u)\cdot (y,v)\to (x+y,u+v+x^{T}Jy)$ where $J$ is a skew-symmetric matrix satisfying a degeneracy condition with $2\le \mathrm{rank}J<d$. Consider the maximal function defined by $$\mathfrak{M}f(x,u)=\underset{t>0}{\sup }|{\int }_{\Sigma }f(x-ty,u-t{x}^{T}Jy)d\mu (y)|,$$ 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 $d\ge 4$, the maximal operator $\mathfrak{M}$ is bounded on $L^p(G)$ for the range $(d-1)/(d-2)<p\le \infty$.

25 pages, no figures