Microlocal analysis of singular measures
In this talk, which is based on a joint work with V. Banica, I will investigate the structure of singular measures from a micro-local perspective. I will introduce a notion of $L^1$-regularity wave front set for scalar and vectorial distributions. Our main result is a proper microlocal characterisation of the support of the singular part of tempered Radon measures and of their polar functions at these points. We deduce a sharp $L^1$ elliptic regularity result which appears to be new even for scalar measures. I will also illustrate the interest of this micro-local approach with a result of propagation of singularities for constrained measures.