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

TL;DR

This work shows that stability of $L^p$ bounds for perturbations of the Nevo–Thangavelu maximal operator, previously known for Métivier groups, fails in general for arbitrary two-step nilpotent groups when the dilation-invariant subspace is tilted. By formulating Hypothesis $\mathscr{H}(r)$ and reducing to tilted submanifolds of $\mathfrak g_1$, the authors derive sharp necessary conditions for $L^p$ boundedness and prove unboundedness results: for $p\le (r+1)/r$ (with $r\ge 2$) and, in the special case $r=1$, for all finite $p$ under mild nondegeneracy of the tilt. The strategy combines a reduction to the $m=1$ model, a careful parabolic scaling analysis, and Nikodym-type geometric constructions to produce localized counterexamples, highlighting that stability under perturbations is highly group-structure dependent. The results illuminate the limits of extending Métivier stability phenomena to broader two-step groups and suggest directions for upper-bound results under additional curvature or degeneracy constraints.

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 $H\neq \mathfrak g_1$ 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 $\mathfrak g_1$.

Theorems & Definitions (3)

• Theorem 1.1
• Proposition 4.1
• proof