Harmonic measure for lower dimensional sets

Mardi 14 novembre 2017 14:00-15:00 - Svitlana Mayboroda - Université du Minnesota

Résumé : Harmonic measure and harmonic functions more generally play a unique role in geometric measure theory : boundedness of the harmonic Riesz transform is equivalent to uniform rectifiability of sets, so is the boundedness of the harmonic square function, to mention only a few results. Unfortunately, the concept of a harmonic measure is intrinsically restricted to co-dimension 1. In this talk, we introduce a new notion of a « harmonic » measure, associated to a suitable degenerate PDE, which serves the higher co-dimensions. We discuss its basic properties and interplay between absolute continuity of our harmonic measure with respect to the Hausdorff measure, square function estimates, and rectifiability of a lower-dimensional set.
This is joint work with Guy David, Max Engelstein, Joseph Feneuil, and Zihui Zhao.

Lieu : Salle 113-115 (Bâtiment 425)

Harmonic measure for lower dimensional sets  Version PDF