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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
Keywords:
Cauchy transforms; analytic capacity
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