2026/07/16 by Roberto Colombo, Xavier Fernández-Real, Xavier Ros-Oton
#math.AP
Given s∈ (0,1) and a bounded Lipschitz domain Ω⊂ ℝn, we establish a quantitative Dahlberg theory for the s-harmonic measure of Ω, ωsx. In the nonlocal setting, the natural reference measure is an integral weight σs in Ωc that behaves like (1-s)dist(⋅, ∂ Ω)-s close to the boundary. Our main result is a scale-invariant reverse-Hölder estimate for the density dωsx/dσs on boundary-centered balls. As a consequence, we obtain L2(Ωc,σs)-solvability of the exterior Dirichlet problem, with estimates for a nonlocal non-tangential maximal function and uniqueness in the natural distributional class. A weighted Gehring argument improves the reverse-Hölder exponent beyond 2 and consequently yields Lq-solvability for a range of exponents extending strictly below 2. Our results apply to general symmetric stable operators comparable to the fractional Laplacian. Moreover, the proofs are compatible with the limit s→ 1- and thus yield the corresponding results for the Laplacian in the nonlocal-to-local limit. The main new step is to convert a fractional Pohozaev identity for the Green function into uniform square estimates on distance level sets of a Lipschitz boundary. As applications, we derive optimal Sobolev regularity estimates for the homogeneous weighted Dirichlet problem and for the inhomogeneous Poisson problem with zero exterior data.