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 $$.