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Zapiski Nauchnykh Seminarov LOMI, 1979, Volume 92, Pages 115–133 (Mi znsl3193)  

This article is cited in 13 scientific papers (total in 13 papers)

On the smoothness of Cauchy type integrals

E. M. Dyn'kin
Abstract: Let $f$ be a continuous function on a closed rectifiable Jordan curve $\Gamma$, $Kf$ its Cauchy integral (Considered as a function analytic in the Jordan region with the boundary $\Gamma$). The article deals with estimates of the smoothness of $Kf$ in $\Bar G$ in terms of moduli of smoothness of $f$. Principal results for the Hölder–Zygmund classes $\bigwedge^\alpha$ are as follows: a) $K[\bigwedge^\alpha(\Gamma)]\subset\bigwedge^\alpha(\Bar G)$ ($\alpha\ge1$); b)$K[\bigwedge^\alpha(\Gamma)]\subset\bigwedge^{2\alpha-1}(\Bar G)$ ($1/2<\alpha<1$); c) there is a $\Gamma$ and an $f\in\bigwedge^\frac12(\Gamma)$ such that $\sup_G|Kf|=+\infty$; d) for every $\beta\in(\max(0,2\alpha-1),d)$ there is a pair ($\Gamma f)$ such that $f\in\bigwedge^\alpha(\Gamma)$, $K[\bigwedge^\alpha(\Gamma)]\subset\bigwedge^\alpha(\Bar G)$, $\omega_G(Kf,\delta)\ge\operatorname{const}\cdot\delta^\beta$ ($\omega_G$ being the continuity modaley). A precise sufficient condition of the continuity of $KF$ in $\Bar G$ (expressed in terms of $\omega_\Gamma(f))$ is given.
Bibliographic databases:
UDC: 517.948:513.8+519.4
Language: Russian
Citation: E. M. Dyn'kin, “On the smoothness of Cauchy type integrals”, Investigations on linear operators and function theory. Part IX, Zap. Nauchn. Sem. LOMI, 92, "Nauka", Leningrad. Otdel., Leningrad, 1979, 115–133
Citation in format AMSBIB
\Bibitem{Dyn79}
\by E.~M.~Dyn'kin
\paper On the smoothness of Cauchy type integrals
\inbook Investigations on linear operators and function theory. Part~IX
\serial Zap. Nauchn. Sem. LOMI
\yr 1979
\vol 92
\pages 115--133
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl3193}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=566745}
\zmath{https://zbmath.org/?q=an:0432.30033}
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  • https://www.mathnet.ru/eng/znsl/v92/p115
  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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