Laplacians on smooth distributions

TL;DR

This paper introduces a Laplacian operator on smooth distributions over compact manifolds, proves its self-adjointness, and shows that functions of this operator act as smoothing operators in the context of singular foliations.

Contribution

It defines a Laplacian on smooth distributions, establishes its self-adjointness, and demonstrates smoothing properties for functions of this Laplacian within singular foliation frameworks.

Findings

The Laplacian $ riangle_H$ is self-adjoint in $L^2(M,)$.

Functions of $ riangle_H$ are smoothing operators on singular foliations.

The proofs utilize pseudodifferential calculus and subelliptic estimates.

Abstract

Let $M$ be a compact smooth manifold equipped with a positive smooth density $\mu$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $\Delta_H$ associated with $(H,\mu,g)$ and prove that it gives rise to an unbounded self-adjoint operator in $L^2(M,\mu)$. Then, assuming that $H$ generates a singular foliation $\mathcal F$, we prove that, for any function $\varphi$ from the Schwartz space $\mathcal S(\mathbb R)$, the operator $\varphi(\Delta_H)$ is a smoothing operator in the scale of longitudinal Sobolev spaces associated with $\mathcal F$. The proofs are based on pseudodifferential calculus on singular foliations developed by Androulidakis and Skandalis and subelliptic estimates for $\Delta_H$.