Laplacians on smooth distributions

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 $\phi$ from the Schwartz space $\mathcal{S}(\mathbb{R})$, the operator $\phi ({\Delta }_H)$ 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 ${\Delta }_H$.

21 pages

References in corpus (3)

• The Egorov theorem for transverse Dirac type operators on foliated manifolds
• Natural Equivariant Dirac Operators
• Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

Cited by in corpus (2)

• Riemannian metrics and Laplacians for generalised smooth distributions
• Laplacians on smooth distributions as $C^{\ast }$-algebra multipliers