Cauchy projectors on non-smooth and non-rectifiable curves

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

Kazan Federal University, 18 Kremlyovskaya str., Kazan 420008, Russia

Аннотация: 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 $H_{\nu}(\Gamma)$, $0<\nu<1$ (considered as countably-normed Frechet space) onto itself.

Ключевые слова: Cauchy projectors, non-smooth curves, non-rectifiable curves.

УДК: 517.544

MSC: 30E20

Поступила в редакцию: 28.07.2018
Исправленный вариант: 24.12.2018
Принята в печать: 21.12.2018

Язык публикации: английский

DOI: 10.15393/j3.art.2019.5870

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