Obstructions to exact regularity of sublaplacians

Abstract

Let $M$ be a closed manifold equipped with a collection of smooth real vector fields $X_1,\ldots, X_k$ and a smooth measure $μ$. Let $P:=\sum_j X_j^\dagger X_j $ be the associated sublaplacian. Under the assumption that each pair of points of $M$ can be connected by a path obtained by joining integral curves of the vector fields, the equation \[ Pu=f \] admits unique zero-average solutions $u$ in the natural energy space, for all zero average data $f\in L^2$. We say that exact regularity holds for the above equation if $u\in H^k(M)$ whenever $f\in H^k(M)$, where $H^k(M)$ is the $L^2$ based Sobolev space of any order $k\in \mathbb{N}$. We investigate how the presence of a characteristic submanifold, namely a submanifold tangent to all vector fields $X_j$, may cause a failure of exact regularity. We prove that this indeed happens for a class of "worm sublaplacians", which are real analogues of Kohn Laplacians on Diederich--Fornaess worm domains, of interest in several complex variables. Our main tool is a scaling lemma inspired by work of Barrett and Christ on the $\bar\partial$-Neumann problem.