# zbMATH — the first resource for mathematics

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:
 3e+21 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
##### Keywords:
Neumann operator; Cauchy operator; chord-arc curves