TY - JOUR
T1 - Harmonic measure and Riesz transform in uniform and general domains
AU - Mourgoglou, Mihalis
AU - Tolsa, Xavier
PY - 2021/1/1
Y1 - 2021/1/1
N2 - Let Ω ⊈ ℝn+1 be open and let μ be some measure supported on ∂Ω such that μ.B(x, r)) ≤ C rn for all x ∊ Rn+1, r > 0. We show that if the harmonic measure in Ω satisfies some scale invariant A1-type conditions with respect to μ, then the n-dimensional Riesz transform (equcation presented) is bounded in L2(μ). We do not assume any doubling condition on μ. We also consider the particular case when Ω is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.
AB - Let Ω ⊈ ℝn+1 be open and let μ be some measure supported on ∂Ω such that μ.B(x, r)) ≤ C rn for all x ∊ Rn+1, r > 0. We show that if the harmonic measure in Ω satisfies some scale invariant A1-type conditions with respect to μ, then the n-dimensional Riesz transform (equcation presented) is bounded in L2(μ). We do not assume any doubling condition on μ. We also consider the particular case when Ω is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.
UR - http://www.scopus.com/inward/record.url?scp=85037705588&partnerID=8YFLogxK
U2 - 10.1515/crelle-2017-0037
DO - 10.1515/crelle-2017-0037
M3 - Article
AN - SCOPUS:85037705588
SN - 0075-4102
VL - 2020
SP - 183
EP - 221
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
IS - 758
ER -