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-H\"{o}lder 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-H\"{o}lder 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.