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Square functions, Cauchy integrals, analytic capacity, and harmonic measure. (English) Zbl 0675.30029
Harmonic analysis and partial differential equations, Proc. Int. Conf., El Escorial/Spain 1987, Lect. Notes Math. 1384, 24-68 (1989).
[For the entire collection see Zbl 0669.00010.]
This paper deals with four topics all of which relate certain geometric properties to \(L^ 2\)-estimates of certain functions. The first two topics are concerned with the \(L^ 2\) boundedness of the Cauchy integral operators; one is a new proof via a geometric lemma of Carleson type in the case of Lipschitz curves, and the other gives an alternative approach to G. David’s proof [Ann. Sci. Éc. Norm. Super., IV. Sér. 17, 157-189 (1984; Zbl 0537.42016)] for Ahlfors-David curves. The third topic is on analytic capacity and the author makes several comments on results given in T. Murai [A real variable method for the Cauchy transform, and analytic capacity (1988; Zbl 0645.30016)]. As the fourth topic, the author presents a geometric condition under which a simply connected domain is a Makarov domain, namely a domain with a boundary set whose harmonic measure is 1 but having zero Hausdorff measure with respect to a certain measure function h with \[ \lim_{t\to 0}[h(t)/t]=\infty. \]
Reviewer: F.-Y.Maeda

MSC:
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30C85 Capacity and harmonic measure in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
44A15 Special integral transforms (Legendre, Hilbert, etc.)