Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$ ,

Abstract

Abstract. We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [RR]. 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 ⊂ R n+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 [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant L q bounds, with q > 1, on the Poisson kernel.Résumé. On présente une version invariante paréchelles et en dimension supé-rieureà 3 d'un théorème classique de F. et M. Riesz [RR]. Plus précisément, oń etablit l'absolue continuité de la mesure harmonique par rapportà la mesure de surface, ainsi qu'un gain d'intégrabilité pour le noyau de Poisson, pour un domaine Ω ⊂ R n+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 [HMU]établit une réciproque, c'est-à-dire l'uniforme rectifiabilité du bord en supposant des estimées invariantes paréchelle L q pour q > 1 sur le noyau de Poisson.

A new characterization of chord-arc domains

“…We observe that (3.7) is a consequence of a Poincaré inequality proved in [HM,Section 4]. Moreover, by [HM,Lemma 5.1,Lemma 5.3, and Corollary 5.6], we have…”

“…We let B * := κ 0 B be a concentric dilate of B with κ 0 a large constant (see [HM,(5.12)]), and set ∆ * := B * ∩ ∂Ω. As in the proof of [HM,Lemma 5.10], we may choose β so that the following properties hold:…”

Rectifiability of harmonic measure

“…To be precise, [Da], [DJ], [Se] establish a quantitative scale-invariant result, the A ∞ property of harmonic measure, which in the planar case was proved by Lavrent'ev [Lv]. See also [HM1], [BL], [Ba], [Az], [AMT1], [Mo], [ABHM] along with [AHMNT] in this context. We shall not give a precise definition of NTA domains here (see [JK]), but let us mention that they necessarily satisfy interior and exterior cork-screw conditions as well as a Harnack chain condition, that is, certain quantitative analogues of connectivity and openness, respectively.…”

“…We refer the reader to [Hel,Chapter 3] for full details. For unbounded domains, harmonic measure can similarly be defined (see for example [HM1,Section 3]). …”

“…Here we will follow the construction of harmonic measure from [HM1,Section 3]. To that end we fix x 0 ∈ ∂Ω and N large enough and set Ω N = Ω ∩ B(x 0 , 2N ), whose harmonic measure is denoted by ω N .…”