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Generalized Neumann-Poincaré operator and chord-arc curves. (English) Zbl 0736.30028
The author begins with a survey of David’s results on the boundedness of the Cauchy operator \((Ch)(z_ 0)={1\over \pi i}P.V.\int_{\Gamma}h(z)(z-z_ 0)^{-1}dz\) in the space of complex valued functions \(h\in L^ p(\Gamma)\), with \(p>1\). If \(h\) is real valued, \(h\in L^ p_ R(\Gamma)\), and \(C\) is split into its real and imaginary parts, \(C=C_ 1+C_ 2\), one obtains the generalized Neumann- Poincaré operator \(C_ 1\). The theme of the paper is the characterisation of Neumann domains and chord-arc curves \(\Gamma\) by properties of the operators \(C_ 1\) and \(C_ 2\) acting on functions \(h\) in \(L^ p_ 0\) (in \(L^ p_ R\) with \(\int_ \Gamma hds=0\)). We mention two results. 1. A Jordan curve, regular in the sense of Ahlfors- David, is chord-arc iff \(C_ 2\) is a bounded isomorphism of \(L^ p_ 0\) for some \(p>1\). 2. If \(\Gamma\) is AD-regular and the norm of \(C_ 1\) with respect to \(L^ p_ 0\) is \(d<1\), for some \(p>1\), then \(\Gamma\) is chord-arc.
Reviewer: D.Gaier (Gießen)

MSC:
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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