Microlocal approach to the Hausdorff dimension of measures

Abstract

In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. We prove our results by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of the Uncertainty Principle for measures on the complex sphere. This instance of the UP generalizes the celebrated theorem of Aleksandrov and Forelli concerning regularity of pluriharmonic measures.