TY - JOUR

T1 - Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem

AU - Azzam, Jonas

AU - Hofmann, Steve

AU - Martell, Jose Maria

AU - Mourgoglou, Mihalis

AU - Tolsa, Xavier

PY - 2020/12/31

Y1 - 2020/12/31

N2 - It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω⊂R^n+1 with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂Ω) for some p<∞. In this paper, we give a geometric characterization of the weak-A∞ property, of harmonic measure, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are in the nature of best possible: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds).

AB - It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω⊂R^n+1 with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂Ω) for some p<∞. In this paper, we give a geometric characterization of the weak-A∞ property, of harmonic measure, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are in the nature of best possible: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds).

U2 - 10.1007/s00222-020-00984-5

DO - 10.1007/s00222-020-00984-5

M3 - Article

VL - 222

SP - 881

EP - 993

JO - Inventiones mathematicae

JF - Inventiones mathematicae

SN - 0020-9910

IS - 3

ER -