Abstract
This paper is mostly a review paper. It contains a description of old and recent results concerning the regularity conditions on a Jordan curve in the plane that imply the boundedness of the singular integral operator as well as the boundary behavior of the Cauchy type integral. These results are of significance for boundary value problems in domains with non-smooth and non-rectifiable boundaries.
Similar content being viewed by others
References
Andrievskiĭ, V.V.: On the question of the smoothness of an integral of Cauchy type. Ukrain. Mat. Zh. (Russian) 38(2), 139–149 (267) (1986)
Bari, N.K., Stečkin, S.B.: Best approximations and differential properties of two conjugate functions. Trudy Moskov. Mat. Obšč (Russian) 5, 483–522 (1956)
Babaev, A.A.: A singular integral with continuous density. Azerbaĭdžan. Gos. Univ. Učen. Zap. Ser. Fiz.-Mat. Nauk (Russian) (5), 11–23 (1965)
Babaev, A.A., Salaev, V.V.: Boundary value problems and singular equations on a rectifiable contour. Mat. Zametki (Russian) 31(4), 571–580 (654) (1982)
Babaev, A.A., Salaev, V.V.: A one-dimensional singular operator with continuous density along a closed curve. Dokl. Akad. Nauk SSSR (Russian) 209, 1257–1260 (1973)
Bory Reyes, J., Abreu Blaya, R.: One-dimensional singular integral equations. Complex Var. Theory Appl. 48(6), 483–493 (2003)
Bustamante, G.J.: Approximation by Lipschitz functions and its application to boundary value of Cauchy-type integrals (English). Approximation and optimization, Proc. Int. Semin., Havana/Cuba 1987. Lect. Notes Math. 1354, 106–110 (1988)
Bustamante, G.J.: The Dini condition and the Cauchy type integral. Rev. Cienc. Mat. (Spanish, English summary) 8(2), 9–14 (1987)
Calderón, A.P.: Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA 74, 1324–1327 (1977)
Calderón, A.P., Calderon, C.P., Fabes, E., Jodeit, M., Rivire, N.M.: Applications of the Cauchy integral on Lipschitz curves. Bull. Am. Math. Soc. 84(2), 287–290 (1978)
Coifman, R.R., McIntosh, A., Meyer, Y.: L’intgrale de Cauchy dfinit un oprateur born sur \(L^{2}\) pour les courbes lipschitziennes (French). [The Cauchy integral defines a bounded operator on \(L^{2}\) for Lipschitz curves]. Ann. Math. (2) 116(2), 361–387 (1982)
Christ, M.A.: \(T(b)\) Theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2), 601–628 (1990)
David, G.: Oprateurs intgraux singuliers sur certaines courbes du plan complexe (French). [Singular integral operators over certain curves in the complex plane]. Ann. Sci. cole Norm. Sup. (4) 17(1), 157–189 (1984)
David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, 38, p. xii+356. American Mathematical Society, Providence (1993)
Davydov, N.A.: The continuity of the Cauchy type integral in a closed region. Dokl. Akad. Nauk. SSSR (Russian) 64(6), 759–762 (1949)
Davydov, N.A.: Certain questions of theory of boundary value of analytical functions. Ph.D. thesis, Moscow (1949)
Dyn’kin, E.M.: Smoothness of Cauchy type integrals. Zap. Nauchn. Sem. Leningr. Otd. Mat in-ta AN SSSR (92), 115–133 (1979)
Dyn’kin, E.M.: On the smoothness of integrals of Cauchy type. Sov. Math., Dokl (Russian, English) 21, 199–202 (1980). (translation from Dokl. Akad. Nauk SSSR 250, 794–797, 1980)
Gakhov, F.D.: Kraevye zadachi (Russian). [Boundary value problems], 3rd edition, p 640. Revised and Augmented Izdat. “Nauka”, Moscow (1977)
Gerus, O.F.: Moduli of smoothness of the Cauchy-type integral on regular curves. J. Nat. Geom. 16(1–2), 49–70 (1999)
Guseĭnov, E.G.: The Plemelj–Privalov theorem for generalized Hlder classes. Mat. Sb. (Russian) 183(2), 21–37 (1992). (translation in Russian Acad. Sci. Sb. Math. 75, 1, 165–182, 1993)
Kats, B.A.: The Riemann problem on a closed Jordan curve. Sov. Math. (English) 27(4), 83–98 (1983)
Kats, B.A.: The Riemann boundary value problem for nonsmooth arcs and fractal dimensions. St. Petersbg. Math. J. (Russian, English) 6(1), 147–171 (1995). (translation from Algebra Anal. 6, No. 1, 172–202, 1994)
Kats, B.A.: The Cauchy integral along \(\Phi \)-rectifiable curves. Lobachevskii J. Math. (English) 7, 15–29 (2000)
Kats, B.A.: On a generalization of a theorem of N. A. Davydov. Izv. Vyssh. Uchebn. Zaved. Mat (Russian) (1), 39–44 (2002) (translation in Russian Math. Iz. VUZ 46, no. 1, 37–42, 2002)
Kats, B.A.: The Cauchy integral over non-rectifiable paths. Contemp. Math. 455, 183–196 (2008)
Kats, B.A.: The Cauchy transform of certain distributions with application. Complex Anal. Oper. Theory 6(6), 1147–1156 (2012)
Kolmogorov, A.N., Tikhomirov, V.M.: \(\varepsilon \)-Entropy and capasity of set in functional spaces. Uspekhi Math. Nauk 14, 3–86 (1959)
Lesniewicz, R., Orlicz, W.: On generalized variations (II). Stud. Math. 45(1), 71–109 (1973)
Lu, J.K.: Boundary Value Problems for Analytic Functions. Series in Pure Mathematics, 16, p. xiv+466. World Scientific Publishing Co., Inc, River Edge (1993)
Magnaradze, L.: On a generalization of the theorem of Plemelj–Privalov. Soobščeniya Akad. Nauk Gruzin. SSR (Russian) 8, 509–516 (1947)
Mattila, P.: Rectifiability, analytic capacity, and singular integrals. Doc. Math. J. DMV Extra Vol. ICM Berlin (English) II, 657–664 (1998)
Mattila, P.: Singular integrals, analytic capacity and rectifiability. J. Fourier Anal. Appl. (English) 3, 797–812 (1997). (Spec. Iss.)
Mattila, P., Melnikov, M.S., Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability (English). Ann. Math. (2) 144(1), 127–136 (1996)
Muskhelishvili, N.I.: Singular Integral Equations (English), 3rd edn, p. 447. Wolters-Noordhoff Publishing, Groningen (1967). (translated from the second Russian edition)
Plemelj, J.: Ein Erganzungssatz zur Cauchy’schen Integraldarstellung analytisher Funktionen. Randwerte betreffend Monatsh. für Math. und Phys. B 19S, 205–210 (1908)
Privaloff, I.: Sur les fonctions conjuguess. Bull. Soc. Math. France 44(2–3), 100–103 (1916)
Privaloff, I.: Sur les integrales du type de Cauchy (French). C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 23, 859–863 (1939)
Privaloff, I.: Graničnye svoĭstva analitičeskih funkciĭ (Russian). [Boundary Properties of Analytic Functions], 2nd edn, p. 336. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad (1950)
Paatashvili, V.A., Khuskivadze, G.A.: Boundedness of a singular Cauchy operator in Lebesgue spaces in the case of nonsmooth contours. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk. Gruzin. SSR 69, 93–107 (1982). (Russian)
Rahimov, R.M.: Behavior of an integral of Cauchy type ”near” the line of integration. Azerbaĭdzhan. Gos. Univ. Uchen. Zap. (Russian) (1), 51–59 (1979)
Salaev, V.V.: Direct and inverse estimates for a singular Cauchy integral along a closed curve (English). Math. Notes 19, 221–231 (1976)
Salaev, V.V., Tokov, A.O.: Necessary and sufficient conditions for the continuity of Cauchy tye integral in closed domain. Doklady Academic Nauk Azer 39(12), 7–11 (1983)
Salaev, V.V., Guseĭnov, E.G., Seĭfullaev, R.K.: The Plemelj–Privalov theorem. Dokl. Akad. Nauk SSSR (Russian) 315(4), 790–793 (1990). (translation in Soviet Math. Dokl. 42, 1991, no. 3, 849–852)
Salimov, T.S.: A Singular Cauchy Integral in \(H_\omega \) Spaces (Russian). Theory of Functions and Approximations, Part 2 (Saratov, 1982), pp.130–134. Saratov. Gos. Univ., Saratov (1983)
Salimov, T.S.: The singular Cauchy integral in spaces \(L_p,\;p\ge 1\). Akad. Nauk Azerbaĭdzhan. SSR Dokl. (Russian) 41(3), 3–5 (1985)
Tamrazov, P.M.: Boundary and solid properties of holomorphic functions in a complex domain. Sov. Math. Dokl. (Russian, English) 13, 725–730 (1972). (translation from Dokl. Akad. Nauk SSSR 204, 565–568, 1972)
Tamrazov, P. M.: Gladkosti i polinomialnye priblizheniya (Russian). [Smoothnesses and Polynomial Approximations], p. 271. Izdat ”Naukova Dumka”, Moscow (1975)
Tolsa, X.: Principal values for the Cauchy integral and rectifiability (English). Proc. Am. Math. Soc. 128(7), 2111–2119 (2000)
Tolsa, X.: \(L^2\)-Boundedness of the Cauchy integral operator for continuous measures (English). Duke Math. J. 98(2), 269–304 (1999)
Zygmund, A.: Trigonometric Series, 2nd edn. Vols. I, II. Cambridge University Press, New York (1959) (Vol. I. xii+383 pp.; Vol. II. vii+354)
Acknowledgments
The authors are very thankful to the anonymous referee for his/her valuable remarks and suggestions that have considerably improved this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
B. Kats is partially supported by Russian Foundation for Basic Researches, grants 12-01-00636-a, 13-01-00322-a and 12-01-97015-r-povolzhie-a.
Rights and permissions
About this article
Cite this article
Blaya, R.A., Reyes, J.B. & Kats, B. Cauchy integral and singular integral operator over closed Jordan curves. Monatsh Math 176, 1–15 (2015). https://doi.org/10.1007/s00605-014-0656-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-014-0656-9