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

Abstract

Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak{g}={\mathfrak{g}}_1\oplus {\mathfrak{g}}_2$, where $[\mathfrak{g},\mathfrak{g}]\subset {\mathfrak{g}}_2$. We consider maximal functions associated to spheres in a $d$-dimensional linear subspace $H$, dilated by the automorphic dilations. $L^p$ boundedness results for the case where $H={\mathfrak{g}}_1$ are well understood. Here we consider the case of a tilted hyperplane $H\ne {\mathfrak{g}}_1$ which is not invariant under the automorphic dilations. In the case of Métivier groups it is known that the $L^p$-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 $L^p$ boundedness. We prove these results in a more general setting with tilted versions of submanifolds of ${\mathfrak{g}}_1$.

Minor revisions. 12 pages