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

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 $2\le \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}J<d$. Consider the maximal function defined by $$\mathfrak{M}f(x,u)=\underset{t>0}{\sup }\mid {\int }_{\mathrm{\Sigma }}f(x-ty,u-t{x}^{T}Jy)d\mu (y)\mid ,$$ where $\mathrm{\Sigma }$ is a smooth convex hypersurface and $d\mu$ is a compactly supported smooth density on $\mathrm{\Sigma }$ such that the Gaussian curvature of $\mathrm{\Sigma }$ 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 $(d-1)\mathrm{/}(d-2)<p\le \mathrm{\infty }$.