About the convolution of distributions on groupoids

TL;DR

This paper develops a convolution product for distributions on Lie groupoids, expanding the algebraic framework and analyzing microlocal properties, wave front sets, and applications to pseudodifferential and Fourier integral operators.

Contribution

It introduces a new convolution algebra of distributions on Lie groupoids and studies its microlocal properties, extending the existing algebra of smooth compactly supported functions.

Findings

Constructed a unital involutive algebra enlarging the convolution algebra

Established conditions on wave front sets for convolution

Connected wave front set behavior to the symplectic groupoid $T^*G$

Abstract

We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra $\cE\_{r,s}'(G,\Omega^{1/2})$ enlarging the convolution algebra $C^\infty\_c(G,\Omega^{1/2})$ associated with any Lie groupoid $G$. We prove that $G$-operators are convolution operators by transversal distributions. We also investigate the microlocal aspects of the convolution product. We give conditions on wave front sets sufficient to compute the convolution product and we show that the wave front set of the convolution product of two distributions is essentially the product of their wave front sets in the symplectic groupoid $T^*G$ of Coste-Dazord-Weinstein. This also leads to a subalgebra $\cE\_{a}'(G,\Omega^{1/2})$ of $\cE\_{r,s}'(G,\Omega^{1/2})$ which contains for instance the algebra of pseudodifferential $G$-operators and a class of Fourier integral $G$-operators which will be the central theme of a forthcoming paper.