Applications of harmonic measure
This monograph illustrates how elementary harmonic measure arguments have broad applications. The author presents some recent results on harmonic measure and applications of harmonic measure estimates to problems in analysis and spectral theory. Most of the results included are not available in any other book. The treatment is elementary in that Brownian motion is not used--the introduction gives all the background needed for following the text. Chapters cover length sums, level curves of conformal mappings, interpolating sequences, nontangential limit sets, Makarov's theorems, and periodic spectra of Hill's equation.
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Level Curves of Conformol Moppings
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Ahlfors analytic annulus APPLICATIONS OF HARMONIC applies assume band lengths Beurling Bloch function Borel Boundary bounded harmonic function Brownian motion proof Brownian paths Carleson closed disc compact support conformal mapping conformally invariant connected domains Const constant define definition l.l dfl(h diam(D Dirichlet problem discontinuous group disjoint dist(w domain fl finite Gaps and Bands Garnett harmonic measure Hausdorff measure Hence hits hyperbolic metric interpolating sequence Jordan curve Jordan domain Kakutani leave fl(h log log log Math maximum principle Mobius transformation n-th gap nontangential limit set one-to-one plane domain Proof of Theorem prove quarter plane regular point Riesz theorem satisfies selfadjoint simply connected slit half disc solution Suppose Theorem 9 Theory Trubowitz Uj(z uniformly unit ball unit disc Univalent Functions University of Arkansas upper half plane vector Vj(z Volume yields zeros of A(X