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 $(d-1)/(d-2)<p\leq\infty$.
Categories: math.CA