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A real variable method for the Cauchy transform, and analytic capacity. (English) Zbl 0645.30016
Lecture Notes in Mathematics, 1307. Berlin etc.: Springer-Verlag. vi, 133 p. DM 23.00 (1988).
The pioneering work of A. G. Vitushkin in the 1960’s showed that analytic capacity \(\gamma\) is a fundamental concept in a number of problems concerning uniform approximation in the complex plane. Therefore there has been much interest in estimates of the set function \(\gamma\) in terms of metric concepts, capacities etc. The purpose of this book is to prove some new results about \(\gamma\) and the basic tool for obtaining these results is the Cauchy (-Hilbert) transform \[ H_{\Gamma}f(z)=\pi^{- 1}p.v.\int_{\Gamma}f(\zeta)(\zeta -z)^{-1}| d\zeta |;\;z\in \Gamma, \] where \(\Gamma\) is a finite union of disjoint smooth arcs. A number of impressive results are presented, both about \(H_{\Gamma}\) and various metric estimates of \(\gamma\). The last theorem of the book, a recent result due to Peter Jones and the author, serves to illustrate this: For \(0<\alpha <1\) define the integral-geometric quantity \(Cr_{\alpha}(\cdot)\) (the \(\alpha\)-Crofton length) as follows: \[ Cr_{\alpha}(E)=\lim_{\epsilon \to 0}(\inf \int^{\pi}_{- \pi}\int^{\infty}_{0}| \partial(\cup_{k}D(z_ k,r_ k))\cap L(r,\theta)|^{\alpha} dr d\theta, \] where L(r,\(\theta)\) is the straight line x cos \(\theta+y \sin\theta =r\), \(| F|\) denotes the number of elements of the set F and the inf is taken over all coverings \(\{D(z_ k,r_ k)\}\) \(n_{k=1}\) of the compact set E with radii less than \(\epsilon\). Then for all \(0<\alpha <\) there exists \(E_{\alpha}\) such that \[ \gamma(E_{\alpha}) = 1\text{ and } Cr_{\alpha}(E_{\alpha})=0 \] (Theorem F).
Reviewer: B.Øksendal

MSC:
30C85 Capacity and harmonic measure in the complex plane
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