The convolution algebra of Schwarz kernels on a singular foliation

Motivated by the study of H\"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse 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, and on the leaves and their holonomy covers. 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.