Cauchy projectors on non-smooth and non-rectifiable curves

B. A. Kats ; S. R. Mironova ; A. Yu. Pogodina

Problemy analiza, Tome 8 (2019) no. 1, pp. 65-71

Résumé

Let $f(t)$ be defined on a closed Jordan curve $\Gamma$ that divides the complex plane on two domains $D^{+}$, $D^{-}$, $\infty\in D^{-}$. Assume that it is representable as a difference $f(t)=F^{+}(t)-F^{-}(t)$, $t\in\Gamma$, where $F^{\pm}(t)$ are limits of a holomorphic in $\overline{\mathbb C}\setminus\Gamma$ function $F(z)$ for $D^{\pm}\ni z\to t\in\Gamma$, $F(\infty)=0$. The mappings $f\mapsto F^{\pm}$ are called Cauchy projectors. Let $H_{\nu}(\Gamma)$ be the space of functions satisfying on $\Gamma$ the Hölder condition with exponent $\nu\in (0,1].$ It is well known that on any smooth (or piecewise-smooth) curve $\Gamma$ the Cauchy projectors map $H_{\nu}(\Gamma)$ onto itself for any $\nu\in (0, 1)$, but for essentially non-smooth curves this proposition is not valid. We will show that even for non-rectifiable curves the Cauchy projectors continuously map the intersection of all spaces $H_{\nu}(\Gamma)$, $0\nu1$ (considered as countably-normed Frechet space) onto itself.

Détail

Keywords: Cauchy projectors, non-smooth curves
Mots-clés : non-rectifiable curves.