Laplacians on smooth distributions

Abstract

Let be a compact smooth manifold equipped with a positive smooth density and let be a smooth distribution endowed with a fibrewise inner product . We define the Laplacian associated with and prove that it gives rise to an unbounded self-adjoint operator in . Then, assuming that generates a singular foliation , we prove that, for any function in the Schwartz space , the operator is a smoothing operator in the scale of longitudinal Sobolev spaces associated with . The proofs are based on pseudodifferential calculus on singular foliations, which was developed by Androulidakis and Skandalis, and on subelliptic estimates for . Bibliography: 35 titles.