Dahlberg's theorem on harmonic measure

Remarks

More precisely, $k^x$ satisfies a reverse Hölder inequality with exponent $2$ on surface balls, and by Gehring's lemma it lies in $L^{2 + \varepsilon}(\sigma)$ for some $\varepsilon > 0$ depending on $\Omega$. As a consequence, the Dirichlet problem on $\Omega$ is solvable for boundary data in $L^p(\sigma)$ with $p \ge 2$, with nontangential convergence to the data and a nontangential maximal function estimate. Idea of the proof: the boundary Harnack principle compares $\omega^x$ of a surface ball with the Green function at a nearby interior point. One proof then uses a Rellich-type integral identity for harmonic functions on Lipschitz domains to get the $L^2$ bound. Example: for a ball, $k^x$ is the Poisson kernel, which is bounded, so $k^x \in L^p$ for every $p$. The exponent $2$ cannot be replaced by a fixed larger one for all Lipschitz domains. In the plane, at the vertex of a sector of opening $\theta \in (\pi, 2\pi)$, the density behaves like $t^{\pi/\theta - 1}$ at distance $t$ from the vertex. It lies in $L^p$ near the vertex only for $p < 1 / (1 - \pi/\theta)$, and this bound tends to $2$ as $\theta \to 2\pi$.

Going deeper

For a student

Dahlberg's theorem made Lipschitz domains the natural setting for harmonic analysis on rough domains. It is the basis of the $L^p$ theory of the Dirichlet and Neumann problems and of layer potentials on Lipschitz boundaries. It shows that harmonic measure is an $A_\infty$ weight with respect to surface measure, a property later studied on much more general domains. In the plane, for a Jordan domain with rectifiable boundary, the F. and M. Riesz theorem already gives mutual absolute continuity with arc length.