Failure of stability of a maximal operator bound for perturbed Nevo–Thangavelu means

Abstract Let be a two‐step nilpotent Lie group, identified via the exponential map with the Lie‐algebra , where . We consider maximal functions associated to spheres in a ‐dimensional linear subspace , dilated by the automorphic dilations. boundedness results for the case where are well understood. Here, we consider the case of a tilted hyperplane which is not invariant under the automorphic dilations. In the case of Métivier groups, it is known that the ‐boundedness results are stable under a small linear tilt. We show that this is generally not the case for other two‐step groups, and provide new necessary conditions for boundedness. We prove these results in a more general setting with tilted versions of submanifolds of .