Microlocal analysis of singular measures

TL;DR

This paper develops a microlocal framework to analyze the structure of singular measures, leading to new regularity results and insights into the behavior of measures constrained by PDEs, with applications to harmonic analysis and geometric measure theory.

Contribution

It introduces an $L^1$-regularity wave front set for distributions and provides a microlocal characterization of singular measure supports, extending previous results and establishing new regularity theorems.

Findings

Established a microlocal characterization of singular measure supports.

Proved a sharp $L^1$ elliptic regularity result for measures and distributions.

Extended constraints on polar functions and propagation of singularities for PDE-constrained measures.

Abstract

The purpose of this article is to investigate the structure of singular measures from a microlocal perspective. Motivated by the result of De Philippis-Rindler [10] and the notions of wave cone of Murat-Tartar [19,20,26,27] and of polarisation set of Denker [9] we introduce a notion of $L^1$-regularity wave 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. The proof is based on De Philippis-Rindler's approach reinforced by microlocal analysis techniques and some extra geometric measure theory arguments. We deduce a sharp $L^1$ elliptic regularity 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. For instance we prove that $ \Psi^0 L^1\cap \mathcal M_{loc}\subseteq L^1_{loc},$ and in particular we obtain $L^1$ elliptic regularity results as $\Delta u\in L^1_{loc}, D^2 u \in \mathcal M_{loc} \Longrightarrow D^2 u\in L^1_{loc}.$ We also deduce several consequences including extensions of the results in [10] giving constraints on the polar function at singular points for measures constrained by a PDE, and of Alberti's rank one theorem. Finally, we also illustrate the interest of this microlocal approach with a result of propagation of singularities for constrained measures.