Harmonic measure for lower dimensional sets

Harmonic measure for lower dimensional sets

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Svitlana Mayborodo, University of Minnesota/IAS
Fine Hall 314

The recent years have seen dramatic breakthroughs in understanding of equivalence between scale-invariant analytic, geometric, and PDE properties of sets. In particular, boundedness of the harmonic Riesz transform was proved to be equivalent to uniform rectifiability, which further yielded necessary and sufficient conditions for absolute continuity of harmonic measure with respect to the Hausdorff measure on the boundary. Unfortunately, the concept of the harmonic measure and more generally harmonic functions is intrinsically restricted to domains with co-dimension 1 boundary. In this talk, we introduce a new notion of a "harmonic" measure, associated to a degenerate linear PDE, which serves lower dimensional sets, and discuss its basic properties, main challenges, and mysterious new features. The underlying analysis can be extended to create a new version of the fractional Laplacian on Ahlfors regular sets.