Harmonic measure and Riesz transform in uniform and general domains

2015/09/28 by Mourgoglou, Mihalis, Tolsa, Xavier · 2 citations
#30C85 #31A15 #31B15 #35J08 #42B20 #42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1509.08386

Abstract

Let Ω\subsetneq\mathbb Rⁿ⁺¹ be open and let μ be some measure supported on ∂Ω such that μ(B(x,r))≤ C rⁿ for all x∈\mathbb Rⁿ⁺¹, r>0. We show that if the harmonic measure in Ω satisfies some scale invariant A_∞ type conditions with respect to μ, then the n-dimensional Riesz transform R_μf(x) = ∫ \fracx-y|x-y|ⁿ⁺¹ f(y) dμ(y) 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.

Cited by

• On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group
• One-sided Rellich inequalities, Regularity problem and uniform rectifiability

Related

• The measures with L²-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions
• The measures with L²-bounded Riesz transform and the Painlevé problem
• The Riesz transform and quantitative rectifiability for general Radon\n measures
• The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions
• The measures with L²-bounded Riesz transform satisfying a subcritical Wolff-type energy condition