The fractional Laplacian in Lipschitz domains: Dahlberg's Theorem and $L^2$-solvability

Abstract

Given $s\in (0,1)$ and a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^n$, we establish a quantitative Dahlberg theory for the $s$-harmonic measure of $\Omega$, ${\omega }_s^x$. In the nonlocal setting, the natural reference measure is an integral weight ${\sigma }_s$ in ${\Omega }^c$ that behaves like $(1-s)\text{dist}(\cdot ,\partial \Omega {)}^{-s}$ close to the boundary. Our main result is a scale-invariant reverse-Hölder estimate for the density $d{\omega }_s^x/d{\sigma }_s$ on boundary-centered balls. As a consequence, we obtain $L^2({\Omega }^c,{\sigma }_s)$-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 $L^q$-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 $s\to 1^{-}$ 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.

39 pages, 3 figures