Several surprising new results about harmonic measure on plane domains have been proved during the last two decades. The most famous of these results are Makarov’s theorems that harmonic measure on any simply connected domain is singular to Hausdorff measure _α for all α > 1 but absolutely continuous to _α for all α < 1. Also surprising was the extension by Jones and Wolff of Makarov’s α > 1 theorem to all plane domains. Further important new results include thework of Carleson, Jones andWolff, and others on harmonicmeasure for complements of Cantor sets; the work by Carleson and Makarov, Bertilsson, Pommerenke, and others on Brennan’s tantalizing conjecture that for univalent functions__|ϕ_|2−pdxdy < ∞ if 4 3 < p < 4; several new geometric conditions that guarantee the existence of angular derivatives; and the Jones square sum characterization of subsets of rectifiable curves and its applications by Bishop and Jones to a variety of problems in function theory. [download]
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