## Harmonic Measure and Uniform Rectifiability

### Mardi 10 avril 2012 15:30-00:00 - Steve Hofmann - University of Missouri

Résumé : We present a higher dimensional, scale-invariant version of the classical theorem of F. and M. Riesz, which established absolute continuity of harmonic measure with respect to arc length measure, for a simply connected domain in the complex plane with a rectifiable boundary. More precisely, for $d\geq 3$, we obtain scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain $\Omega\subset\mathbbR^d$, with a uniformly rectifiable boundary, which satisfies the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. We also prove the converse, that is, we deduce uniform rectifiability of the boundary, assuming scale invariant $L^p$ bounds, with $p>1$, for the Poisson kernel.Joint work with J. M. Martell, and with Martell and I. Uriarte-Tuero.

Lieu : 425 - 117-119

Harmonic Measure and Uniform Rectifiability  Version PDF
janvier 2021 :
 Département de Mathématiques Bâtiment 307 Faculté des Sciences d'Orsay Université Paris-Saclay F-91405 Orsay Cedex Tél. : +33 (0) 1-69-15-79-56 Département Fermeture du département Laboratoire Formation