Laplacians on smooth distributions

Abstract

Let $M$ be a compact smooth manifold equipped with a positive smooth density $μ$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $Δ_H$ associated with $(H,μ,g)$ and prove that it gives rise to an unbounded self-adjoint operator in $L^2(M,μ)$. Then, assuming that $H$ generates a singular foliation $\mathcal F$, we prove that, for any function $φ$ from the Schwartz space $\mathcal S(\mathbb R)$, the operator $φ(Δ_H)$ 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 $Δ_H$.