Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in Lp
[Uniforme rectifiabilité et mesure harmonique I: l'uniforme rectifiabilité entraîne le noyau de Poisson dans Lp ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 3, pp. 577-654.

On présente une version invariante par échelles et en dimension supérieure à 3, d'un théorème classique de F. et M. Riesz [37]. Plus précisément, on établit l'absolue continuité de la mesure harmonique par rapport à la mesure de surface, ainsi qu'un gain d'intégrabilité pour le noyau de Poisson, pour un domaine Ω⊂ℝn+1,n≥2, à bord uniformément rectifiable, vérifiant une condition de chaîne de Harnack et une condition de type « points d'ancrage » ou « Corkscrew » intérieure (mais pas extérieure). L'article associé [28] établit une réciproque, c'est-à-dire l'uniforme rectifiabilité du bord en supposant des estimées invariantes par échelle Lq pour q>1 sur le noyau de Poisson.

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [37]. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain Ω⊂ℝn+1,n≥2, with a uniformly rectifiable boundary, which satisfies the Harnack chain condition plus an interior (but not exterior) Corkscrew condition. In a companion paper to this one [28], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant Lq bounds, with q>1, on the Poisson kernel.

Publié le :
DOI : 10.24033/asens.2223
Classification : 31B05, 35J08, 35J25, 42B99, 42B25, 42B37
Keywords: Harmonic measure, Poisson kernel, uniform rectifiability, Carleson measures, $A_\infty $ Muckenhoupt weights.
Mot clés : Mesure harmonique, noyau de Poisson, uniforme rectifiabilité, mesures de Carleson, poids de Muckenhoupt $A_\infty $. 
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     title = {Uniform rectifiability and harmonic measure {I:} {Uniform} rectifiability implies {Poisson} kernels in~$L^p$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {577--654},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Hofmann, Steve; Martell, José María. Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 3, pp. 577-654. doi : 10.24033/asens.2223. http://www.numdam.org/articles/10.24033/asens.2223/

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