HARMONIC MEASURE OF CURVES IN THE DISK

DONALD E. MARSHALL CARL SUNDBERG

Abstract. A powerful tool for studying the growth of analytic and harmonic functions is Hall’s Lemma, which states that there is a constant C > 0 so that the harmonic measure of a subset E of the closed unit disk \overline{𝔻} evaluated at 0 satisfies

ω(0, E, 𝔻 \ E) ≥ Cω(0, E_{rad}, 𝔻),  (∗)

where E_{rad} is the radial projection of E onto ∂𝔻. FitzGerald, Rodin and Warschawski proved that if E is a continuum in \overline{𝔻} whose radial projection has length at most π then (*) is true with C = 1, and they asked how large the length, |E_{rad}|, can be in order for their result to be valid. We prove that (*) holds with C = 1 provided |E_{rad}| ≤ θ_{c} ≃ 2π (\frac{350}{360}) and θ_{c} cannot be replaced by a larger number. Fuchs asked for the largest constant C so that (*) holds for all E. We show that for every continuum E ⊂ \overline{𝔻}, (*) holds with C = C_{2π} ≃ .977126698498665669 …, where C_{2π} is the harmonic measure of the two long sides of a 3:1 rectangle evaluated at the center. There are Jordan curves for which equality holds in (*) with C = C_{2π}.