Microlocal analysis of singular measures – Nicolas Burq (Orsay)


Sala Riunioni (Dip. Matematica).


In this talk, I will investigate the structure of singular measures from a microlocal perspective. I will introduce a notion of $L^1$- regularitywave front set for scalar and vector 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. I will deduce a sharp $L^1$-ellipticregularity result which appears to be new even for scalar measures and which enlightens the interest of the techniques from geometric measure theory to the study of harmonic analysis questions. I will also deduce several consequences including extensions of results by De Philippis and Rindler giving constraints on the polar function at singular points for measures constrained by a PDE. Finally, I will also illustrate the interest of this microlocal approach with a result of propagation of singularities for constrained measures. This is based on a joint work with V. Banica (Sorbonne Université).

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