Exposés de recherche

Collection Exposés de recherche

00:00:00 / 00:00:00
335 380

The weak-$A_\infty$ condition for harmonic measure

De Xavier Tolsa

Apparaît également dans la collection : Harmonic analysis of elliptic and parabolic partial differential equations / Analyse harmonique des équations aux dérivées partielles elliptiques et paraboliques

The weak-$A_\infty$ condition is a variant of the usual $A_\infty$ condition which does not require any doubling assumption on the weights. A few years ago Hofmann and Le showed that, for an open set $\Omega\subset \mathbb{R}^{n+1}$ with $n$-AD-regular boundary, the BMO-solvability of the Dirichlet problem for the Laplace equation is equivalent to the fact that the harmonic measure satisfies the weak-$A_\infty$ condition. Aiming for a geometric description of the open sets whose associated harmonic measure satisfies the weak-$A_\infty$ condition, Hofmann and Martell showed in 2017 that if $\partial\Omega$ is uniformly $n$-rectifiable and a suitable connectivity condition holds (the so-called weak local John condition), then the harmonic measure satisfies the weak-$A_\infty$ condition, and they conjectured that the converse implication also holds. In this talk I will discuss a recent work by Azzam, Mourgoglou and myself which completes the proof of the Hofman-Martell conjecture, by showing that the weak-$A_\infty$ condition for harmonic measure implies the weak local John condition.

Informations sur la vidéo

Données de citation

  • DOI 10.24350/CIRM.V.19398003
  • Citer cette vidéo Tolsa, Xavier (24/04/2018). The weak-$A_\infty$ condition for harmonic measure. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19398003
  • URL https://dx.doi.org/10.24350/CIRM.V.19398003

Bibliographie

  • Azzam, J., Mourgoglou, M., & Tolsa, X. (2018). A geometric characterization of the weak-$A_\infty$ condition for harmonic measure. <arXiv:1803.07975> - https://arxiv.org/abs/1803.07975

Dernières questions liées sur MathOverflow

Pour poser une question, votre compte Carmin.tv doit être connecté à mathoverflow

Poser une question sur MathOverflow




Inscrivez-vous

  • Mettez des vidéos en favori
  • Ajoutez des vidéos à regarder plus tard &
    conservez votre historique de consultation
  • Commentez avec la communauté
    scientifique
  • Recevez des notifications de mise à jour
    de vos sujets favoris
Donner son avis