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On the Neumann-Poincaré operator. (English) Zbl 0956.30018
Let \(\Gamma \) be a Jordan curve in the complex plane regular in the sense of Ahlfors and David (AD-regular curve). The authors prove the following characterization of the circle in the class of all AD-regular curves. \(\Gamma \) is the circle iff the Neumann-Poincaré operator (i.e. the double layer potential with the density \(f\) on \(\Gamma \)) transforms the space \(L^2_0(\Gamma)\) of all real valued square integrable functions \(f\) on \(\Gamma \) fulfilling \(\int _{\Gamma }f(\xi)|d\xi |=0\) into itself. This theorem contains a.o. the negative answer to a problem of J. Krzyź from: Linear and complex analysis problem book 3. Part 1. Lecture Notes in Math. 1573. V. P. Havin and N. K. Nikolski (eds.), p. 418 (1994; Zbl 0893.30036).
Reviewer: J.Fuka (Praha)

MSC:
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
47B38 Linear operators on function spaces (general)
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References:
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