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Keywords:
Cauchy’s singular operator; the Neumann-Poincaré operator; curves regular in the sense of Ahlfors and David
Cauchy’s singular operator; the Neumann-Poincaré operator; curves regular in the sense of Ahlfors and David
Summary:
Let $\Gamma $ be a rectifiable Jordan curve in the finite complex plane $\mathbb C$ which is regular in the sense of Ahlfors and David. Denote by $L^2_C (\Gamma )$ the space of all complex-valued functions on $\Gamma $ which are square integrable w.r. to the arc-length on $\Gamma $. Let $L^2(\Gamma )$ stand for the space of all real-valued functions in $L^2_C (\Gamma )$ and put \[ L^2_0 (\Gamma ) = \lbrace h \in L^2 (\Gamma )\; \int _{\Gamma } h(\zeta ) |\mathrm{d}\zeta | =0\rbrace . \] Since the Cauchy singular operator is bounded on $L^2_C (\Gamma )$, the Neumann-Poincaré operator $C_1^{\Gamma }$ sending each $h \in L^2 (\Gamma )$ into \[ C_1^{\Gamma } h(\zeta _0) := \Re (\pi \mathrm{i})^{-1} \mathop {\mathrm P. V.}\int _{\Gamma } \frac{h(\zeta )}{\zeta -\zeta _0} \mathrm{d}\zeta , \quad \zeta _0 \in \Gamma , \] is bounded on $L^2(\Gamma )$. We show that the inclusion \[ C_1^{\Gamma } (L^2_0 (\Gamma )) \subset L^2_0 (\Gamma ) \] characterizes the circle in the class of all $AD$-regular Jordan curves $\Gamma $.
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