Motivated by the study of Hörmander's sums-of-squares operators and their generalizations, we define the convolution algebra of proper distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal.

In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation C*-algebra. To this end, we give the construction of a parametrix.

We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Laplacian in the longitudinal pseudodifferential calculus of the smallest singular foliation which includes the distribution, we prove hypoellipticity.