Spherical maximal functions on two step nilpotent Lie groups

Consider $\mathbb R^d\times \mathbb R^m$ with the group structure of a two-step nilpotent Lie group and natural parabolic dilations. The maximal function originally introduced by Nevo and Thangavelu in the setting of the Heisenberg group deals with noncommutative convolutions associated to measures on spheres or generalized spheres in $\mathbb R^d$. We drop the nondegeneracy condition in the known results on M\'etivier groups and prove the sharp $L^p$ boundedness result for all two step nilpotent Lie groups with $d\ge 3$.