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 \[ \lim_{t\to 0}[h(t)/t]=\infty. \]

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.) |