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 Mat. Zametki, 1976, Volume 19, Issue 3, Pages 365–380 (Mi mz7755)

Direct and inverse estimates for a singular Cauchy integral along a closed curve

V. V. Salaev

Azerbaidzhan State University

Abstract: A new metric characteristic $\theta(\delta)$ of rectifiable Jordan curves is introduced. We will find an estimate of the type of the Zygmund estimate for an arbitrary rectifiable closed Jordan curve in its terms. It is shown that the Plemel'–Privalov theorem on the invariance of Holder's spaces is true for the class of curves satisfying the condition $\theta(\delta)\sim\delta$, which is much wider than the class of piecewise smooth curves (the presence of cusps is admissible). The Bari–Stechkin theorem on the necessary conditions of action of a singular operator in the spaces $H_\omega$ is generalized. It is shown that this theorem is valid for every curve which has a continuous tangent at least at one point and $\theta(\delta)\sim\delta$.

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English version:
Mathematical Notes, 1976, 19:3, 221–231

Bibliographic databases:

UDC: 517.5

Citation: V. V. Salaev, “Direct and inverse estimates for a singular Cauchy integral along a closed curve”, Mat. Zametki, 19:3 (1976), 365–380; Math. Notes, 19:3 (1976), 221–231

Citation in format AMSBIB
\Bibitem{Sal76} \by V.~V.~Salaev \paper Direct and inverse estimates for a singular Cauchy integral along a~closed curve \jour Mat. Zametki \yr 1976 \vol 19 \issue 3 \pages 365--380 \mathnet{http://mi.mathnet.ru/mz7755} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=410234} \zmath{https://zbmath.org/?q=an:0351.44006|0345.44006} \transl \jour Math. Notes \yr 1976 \vol 19 \issue 3 \pages 221--231 \crossref{https://doi.org/10.1007/BF01437855} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. R. K. Seifullaev, “The Riemann boundary value problem on a nonsmooth open curve”, Math. USSR-Sb., 40:2 (1981), 135–148
2. E. G. Guseinov, “Singular integrals in spaces of functions summable with a monotone weight”, Math. USSR-Sb., 60:1 (1988), 29–46
3. T. S. Salimov, “The $A$-integral and boundary values of analytic functions”, Math. USSR-Sb., 64:1 (1989), 23–39
4. E. G. Guseinov, “The Plemelj–Privalov theorem for generalized Hölder classes”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 165–182
5. S. A. Plaksa, V. S. Shpakivskyi, “Limiting values of the Cauchy type integral in a three-dimensional harmonic algebra”, Eurasian Math. J., 3:2 (2012), 120–128
6. Dzh. I. Mamedkhanov, “O neravenstvakh raznykh metrik tipa S. M. Nikolskogo”, Tr. IMM UrO RAN, 18, no. 4, 2012, 240–248
7. R. M. Rzaev, “Properties of singular integrals in terms of maximal functions measuring smoothness”, Eurasian Math. J., 4:3 (2013), 107–119
8. J. I. Mamedkhanov, “The Problem of Approximation in Mean on Arcs in the Complex Plane”, Math. Notes, 99:5 (2016), 697–710
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