Harmonic Measure - Properties

Properties

• For any Borel subset E of ∂D, the harmonic measure ω(x, D)(E) is equal to the value at x of the solution to the Dirichlet problem with boundary data equal to the indicator function of E.

• For fixed D and E ⊆ ∂D, ω(x, D)(E) is an harmonic function of x ∈ D and

Hence, for each x and D, ω(x, D) is a probability measure on ∂D.

• If ω(x, D)(E) = 0 at even a single point x of D, then yω(y, D)(E) is identically zero, in which case E is said to be a set of harmonic measure zero. This is a consequence of Harnack's inequality.

Since explicit formulas for harmonic measure are not typically available, we are interested in determining conditions which guarantee a set has harmonic measure zero.

• F. and M. Riesz Theorem: If is a simply connected planar domain bounded by a rectifiable curve (i.e. if ), then harmonic measure is mutually absolutely continuous with respect to arc length: for all, if and only if .

• Makarov's theorem: Let be a simply connected planar domain. If and for some, then . Moreover, harmonic measure on D is mutually singular with respect to t-dimensional Hausdorff measure for all t>1.

• Dahlberg's theorem: If is a bounded Lipschitz domain, then harmonic measure and (n-1)-dimensional Hausdorff measure are mutually absolutely continuous: for all, if and only if .