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\begin{document}$$G\cong {\mathbb {R}}^{d} \ltimes {\mathbb {R}}$$\end{document} be a finite-dimensional two-step nilpotent group with the group multiplication (x,u)·(y,v)→(x+y,u+v+xTJy)\documentclass[12pt]{minimal}
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\begin{document}$$(x,u)\cdot (y,v)\rightarrow (x+y,u+v+x^{T}Jy)$$\end{document} where J is a skew-symmetric matrix satisfying a degeneracy condition with 2≤rankJ<d\documentclass[12pt]{minimal}
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\begin{document}$$2\le \textrm{rank}\, J <d$$\end{document}. Consider the maximal function defined by Mf(x,u)=supt>0|∫Σf(x-ty,u-txTJy)dμ(y)|,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathfrak {M}}f(x, u)=\sup _{t>0}\big |\int _{\Sigma } f(x-ty, u- t x^{T}Jy) d\mu (y)\big |, \end{aligned}$$\end{document}where Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is a smooth convex hypersurface and dμ\documentclass[12pt]{minimal}
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\begin{document}$$d\mu $$\end{document} is a compactly supported smooth density on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} such that the Gaussian curvature of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is nonvanishing on suppdμ\documentclass[12pt]{minimal}
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\begin{document}$$\textrm{supp}{}d\mu $$\end{document}. In this paper we prove that when d≥4\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 4$$\end{document}, the maximal operator M\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {M}}$$\end{document} is bounded on Lp(G)\documentclass[12pt]{minimal}
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\begin{document}$$L^{p}(G)$$\end{document} for the range (d-1)/(d-2)<p≤∞\documentclass[12pt]{minimal}
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\begin{document}$$(d-1)/(d-2)<p\le \infty $$\end{document}.
