# Article

Full entry | PDF   (0.3 MB)
Keywords:
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 $\Gamma$.
References:
[1] M. G. Arsove: Continuous potentials and linear mass distributions. SIAM Rev. 2 (1960), 177–184. DOI 10.1137/1002039 | MR 0143926 | Zbl 0094.08005
[2] G. David: Opérateurs intégraux singuliers sur certaines courbes du plane complexe. Ann. Scient. Éc. Nor. Sup. 17 (1984), 157–189. MR 0744071
[3] P.L. Duren: Theory of $H^p$ spaces. Academic Press, 1970. MR 0268655
[4] D. Gaier: Integralgleichungen erster Art and konforme Abbildung. Math. Z. 147 (1976), 113–129. DOI 10.1007/BF01164277 | MR 0396926
[5] J. Král, I. Netuka, J. Veselý: Teorie potenciálu II. Státní ped. nakl., Praha, 1972.
[6] J. G. Krzy.z: Some remarks concerning the Cauchy operator on AD-regular curves. Annales Un. Mariae Curie-Skłodowska XLII, 7 (1988), 53–58. MR 1074844 | Zbl 0716.30034
[7] J. G. Krzy.z: Generalized Neumann–Poincaré operator and chord-arc curves. Annales Un. Mariae Curie-Skłodowska XLIII, 7 (1989), 69–78. MR 1158099 | Zbl 0736.30028
[8] J. G. Krzy.z: Chord-arc curves and generalized Neumann-Poincaré operator $C_1^{\Gamma }$. “Linear and Complex Analysis Problem Book 3”, Lecture Notes in Math. 1579, V. P. Havin and N. K. Nikolski (eds.), 1994, p. 418.
[9] E. Martensen: Eine Integralgleichung für die logarithmische Gleichgewichtsbelegung und die Krümmung der Randkurve eines ebenen Gebiets. Z. angew. Math.-Mech. 72 (6) (1992), T596–T599. MR 1178329
[10] Ch. Pommerenke: Boundary behaviour of conformal maps. Springer-Verlag, 1992. MR 1217706 | Zbl 0762.30001
[11] I. I. Priwalow: Randeigenschaften analytischer Funktionen. Translated from Russian, Deutscher Verlag der Wissenschaften, Berlin, 1956. MR 0083565 | Zbl 0073.06501
[12] S. Saks: Theory of the integral. Dover Publications, New York, 1964. MR 0167578

Partner of