On local properties of singular integral

  • J. I. Mamedkhanov Baku State University, Azerbaijan
  • S. Z. Jafarov Muş Alparslan University, Turkey and Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, Baku
Keywords: Regular curve, singular integral, H\H, local class of functios, Plemelj-Privalov theorem


UDC 517.5

Let $\gamma$ be a regular curve. We study the local properties of singular integrals in the $H_{\alpha }^{\alpha +\beta }(t_{0},\gamma)$ class of functions. We obtain a strengthening of the Plemelj\–Privalov theorem for functions from the class $H_{\alpha }^{\alpha +\beta}(t_{0},\gamma).$ It is proved that, at the point $t_{0},$ of increased smoothness for $\alpha +\beta < 1,$ there is only a logarithmic loss.


V. V. Andrievskii, On the question of the smoothness of an integral of Cauchy type, Ukr. Mat. Zh., 38, № 2, 139–149 (1986). DOI: https://doi.org/10.1007/BF01058465

A. A. Babaev, A singular integral with continuous density} (in Russian), Azerb. Gos. Univ. Uch. Zap. Ser. Fiz.-Mat. Nauk, 5, 11–23 (1965).

A. A. Babaev, V. V. Salaev, A one-dimensional singular operator with continuous density along a closed curve} (in Russian), Dokl. Akad. Nauk SSSR, 209, 1257–1260 (1973).

R. A. Blaya, J. B. Reyes, B. Kats, Cauchy integral and singular integral operatorv over closed Jordan curves, Monatsh. Math., 176, 1–15 (2015). DOI: https://doi.org/10.1007/s00605-014-0656-9

E. M. Dyn'kin, On the smoothness of integrals of Cauchy type, Sov. Mat. Dokl., 21, 199–202 (1980);

translation from Dokl. Akad. Nauk SSSR, 250, 794–797 (1980).

E. M. Dyn'kin, Smoothness of Cauchy type integrals} (in Russian), Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. AN SSSR, 92, 115–133 (1979).

N. A. Davydov, The continuity of the Cauchy type integral in a closed region, Dokl. Akad. Nauk SSSR, 64, № 6, 759–762 (1949).

G. David, Operateurs integraux singulers sur certaines courbes du plan complexe, Ann. Sci. Écol. Norm. Supér., 4, 157–189 (1984). DOI: https://doi.org/10.24033/asens.1469

P. L. Duren, Theory of $H^{p}$-spaces, Academic Press (1970).

G. Freid, An approximation theoretical study of the structure of real functions, Stud. Sci. Math. Hung., 5, 141–150 (1970).

G. M. Goluzin, Geometric theory of functions of a complex variable, Trasl. Math. Monogr., 26, AMS, Providence, RI (1968). DOI: https://doi.org/10.1090/mmono/026

O. F. Gerus, Moduli of smoothness of the Cauchy-type integral on regular curves, J. Nat. Geom., 16, № 1-2, 49–70 (1999).

E. G. Guseinov, Plemelj–Privalov theorem for generalized Hölder classes (in~Russian), Mat. Sb., 183, № 2, 21–37 (1992).

V. P. Havin, Continuity in $L_{p} $ of an integral operator with the Cauchy kernel, Vestn. Leningrad Univ., 22, № 7 (1967).

A. Yu. Karlovich, I. M. Spitkovsky, The Cauchy singular operator on weighted variable Lebesgue spaces, Oper. Theory: Adv. and Appl., 236, 275–291 (1992). DOI: https://doi.org/10.1007/978-3-0348-0648-0_17

V. M. Kokilashvili, V. Paatasvili, S. Samko, Boundary value problems for analytic functions in the class of Cauchy type integrals with density in $L^{p(.)}(Gamma)$, Bound. Value Probl., 2005, 43–71 (2005). DOI: https://doi.org/10.1155/BVP.2005.43

V. M. Kokilashvili, S. Samko, Weighted boundedness in Lebesgue spaces with variable exponentsof classical operators on Carleson curves, Proc. A. Razmadze Math. Inst., 138, 106–110 (2005).

V. Kokilashvili, S. G. Samko, Singular integrals weighted Lebesgue spaces with variable exponent, Georgian Math. J., 10, № 1, 145–156 (2003). DOI: https://doi.org/10.1515/GMJ.2003.145

V. Kokilashvili, S. Samko, Singular integrals and potentials in some Banach function spaces with variable exponent, J. Funct. Spaces, 1, № 1, 45–59 (2003). DOI: https://doi.org/10.1155/2003/932158

V. M. Kokilashvili, Boundedness criteria for singular integrals in weighted grand Lebesgue spaces, J. Math. Sci., 170, № 3, 20–33 (2010). DOI: https://doi.org/10.1007/s10958-010-0076-x

V. M. Kokilashvili, A. Meskhi, Boundedness of maximal and singular operators in Morrey spaces with variable exponent, Armen. J. Math., 1, № 1, 18–28 (2008).

B. A. Kats, The Cauchy integral along $Phi$-rectifiable curves, Lobachevskii J. Math., 7, 15–29 (2000).

B. A. Kats, On a generalization of a theorem of N. A. Davydov, Izv. Vyssh. Uchebn. Zaved. Mat., 1, 39–44 (2002); English translation: Russian Math. (Izv. VUZ), 46, № 1, 37–42 (2002).

B. A. Kats, The Cauchy integral over non-rectifiable pats, Contemp. Math., 455, 183–196 (2008). DOI: https://doi.org/10.1090/conm/455/08854

B. A. Kats, The Cauchy transform of certain distributions with application, Complex Anal. and Oper. Theory, 6, № 6, 1147–1156 (2012). DOI: https://doi.org/10.1007/s11785-010-0111-4

L. Magnaradze, On a generalization of the theorem of Plemelj–Privalov (in Russian), Soobshch. Akad. Nauk Gruzin. SSR, 8, 509–516 (1947).

J. I. Mamedkhanov, V. V. Salaev, O new classes of functions related to the local structure of singular integrals and some approximation in them, Abstr. All-Union Symp. Approx. Theory, 91–92 (1980).

J. I. Mamedkhanov, I. B. Dadashova, On one theorem of G. Freud, Ukr. Math. J., 70, № 1, 1578–1584 (2019). DOI: https://doi.org/10.1007/s11253-019-01610-1

N. I. Muskhelishvili, Singular integral equations, 3rd ed., Wolters-Noordhoff Publ., Cöningen (1967).

P. Mattila, Rectifiability, analytic capacity, and singular integral, Doc. Math. J. DMV Extra Vol. ICM Berlin II, 509–516 (1998).

P. Mattila, Singular integrals, analytic capacity and rectifiability, J. Fourier Anal. and Appl., 3, 97–812 (1997). DOI: https://doi.org/10.1007/BF02656486

P. Mattila, M. S. Melnikov, J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. Math. (2), 144, № 1, 127–136 (1996). DOI: https://doi.org/10.2307/2118585

J. Plemelj, Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend,} Monatsh. Math. und Phys., 19, 205–210 (1908). DOI: https://doi.org/10.1007/BF01736696

I. Privaloff, Sur les fonctions conjuguees, Bull. Soc. Math. France, 44, № 2-3, 100–103 (1916). DOI: https://doi.org/10.24033/bsmf.965

I. Privaloff, Sur les integrales du type de Cauchy, Dokl. Akad. Nauk URSS, 23, 859–863 (1939).

V. A. Paatashvili, G. A. Khuskivadze, Boundedness of a singular Cauchy operator in Lebesgue spaces in the case of nonsmooth contours (in Russian), Trudy Tbil. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 69, 93–107 (1982).

J. B. Reyes, R. A. Blaya, One-dimensionel singular equations, Complex Var. Theory and Appl., 48, № 6, 483–493 (2003). DOI: https://doi.org/10.1080/0278107032000077033

R. M. Rahimov, Behavior of an integral of Cauchy type ``near'' the line of integration, Azerb. Gos. Univ. Uchen. Zap., 1, 51–59 (1979).

M. Sallay, Über eine lokale Variante des Priwalowschen Satzes, Stud. Sci. Math. Hung., 6, 427–429 (1971).

V. V. Salaev, Direct and inverse estimates for a singular Cauchy integral along a closed curve, Math. Notes, 19, № 3, 221–231 (1976). DOI: https://doi.org/10.1007/BF01437855

V. V. Salaev, A. O. Tokov, Necessary and sufficient conditions for the continuity of Cauchy type integral in closed domain, Dokl. Akad. Nauk Azerb. SSR, 39, № 12, 7–11 (1983).

V. V. Salaev, E. G. Guseinov, R. K. Seifullaev, The Plemelj–Privalov theorem, Dokl. Akad. Nauk SSSR, 315, № 4, 790–793 (1990); English translation: Soviet Math. Dokl., 42, № 3, 849–852 (1991).

T. S. Salimov, A singular Cauchy integral in $H_{omega}$ spaces, Theory of Functions and Approximations, Pt 2, Saratov (1982), p. 130–134.

T. S. Salimov, The singular Cauchy integral in space $L_{p},$ $pgeq 1$} (in Russian), Dokl. Akad. Nauk Azerb. SSR, 41, № 3, 3–5 (1985).

S. G. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct., 16, № 5-6, 461–482 (2005). DOI: https://doi.org/10.1080/10652460412331320322

How to Cite
Mamedkhanov, J. I., and S. Z. Jafarov. “On Local Properties of Singular Integral”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 5, May 2023, pp. 614 -27, doi:10.37863/umzh.v75i5.6959.
Research articles