Mutual absolute continuity of harmonic measures

Remarks

Here $\omega^x(E)$ is the value at $x$ of the Perron–Wiener–Brelot solution of the Dirichlet problem with boundary data the indicator function of $E$. Idea of the proof: for a fixed Borel set $E$, the function $z \mapsto \omega^z(E)$ is harmonic on $\Omega$ with values in $[0, 1]$. Join $x$ to $y$ by a path in $\Omega$ and cover it by finitely many balls whose doubles lie in $\Omega$. Harnack's inequality on each ball compares the values of a nonnegative harmonic function at nearby points. The chain of balls then gives a constant $C$ that does not depend on $E$. Example: for the unit disc, $d\omega^z = \frac{1 - \abs{z}^2}{\abs{e^{i\theta} - z}^2} \frac{d\theta}{2\pi}$. With $y = 0$ and $\abs{x} = r$, the density lies between $\frac{1 - r}{1 + r}$ and $\frac{1 + r}{1 - r}$, so one can take $C = \frac{1 + r}{1 - r}$. Connectedness is needed: on a disjoint union of two discs, the harmonic measures at points of different discs are carried by disjoint circles.

Going deeper

For a student

Because of this result one speaks of sets of harmonic measure zero without naming the starting point. This is how the class of null sets of harmonic measure enters statements such as Fatou-type theorems, Dahlberg's theorem and Makarov's theorem. The constant $C$ blows up as $x$ or $y$ approaches the boundary, which is why boundary estimates need finer tools such as the boundary Harnack principle.