Ukrainian Mathematical Journal

, Volume 60, Issue 11, pp 1683–1690 | Cite as

A property of the β-Cauchy-type integral with continuous density

  • R. Abreu Blaya
  • J. Bory ReyesEmail author

A theorem from the classical complex analysis proved by Davydov in 1949 is extended to the theory of solution of a special case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator \(\partial _{{\overline{z} }} - {\upbeta} \frac{Z}{{\overline{Z} }}\partial _{z} ,\;0 \leq {\upbeta} < 1\)).

It is proved that if γ is a rectifiable Jordan closed curve and f is a continuous complex-valued function on γ such that the integral
$${\int\limits_{{\upgamma}\backslash {\left\{ {{\upzeta} \in {\upgamma} :{\left| {{\upzeta} - t} \right|} \leq r} \right\}}} {\frac{{{\left| {f{\left( {\upzeta} \right)} - f{\left( t \right)}} \right|}}}{{{\left| {{\upzeta} - t{\left| {t \mathord{\left/ {\vphantom {t {\upzeta}}} \right. \kern-\nulldelimiterspace} {\upzeta}} \right|}^{{\uptheta}} } \right|}}}{\left| {{\left( {n{\left( {\upzeta} \right)} - {\upbeta} \frac{{\upzeta}}{{\overline{{\upzeta}} }}\overline{n} {\left( {\upzeta} \right)}} \right)}} \right|}ds,\quad {\uptheta} = \frac{{2{\upbeta}}}{{1 - {\upbeta}}},} }$$
converges uniformly on γ as r → 0, where n(ζ) is the unit vector of outer normal on γ at a point ζ and ds is the differential of arc length, then the β-Cauchy-type integral
$$\frac{1}{{2{\left( {1 - {\upbeta} } \right)}{\uppi} }}{\int\limits_{\mathrm \gamma} {\frac{{f{\left( {\upzeta} \right)}}}{{{\upzeta} - z{\left| {z \mathord{\left/ {\vphantom {z {\upzeta} }} \right. \kern-\nulldelimiterspace} {\upzeta} } \right|}^{\uptheta } }}{\left( {n{\left( {\upzeta} \right)} - {\upbeta} \frac{{\upzeta} }{{\overline{{\upzeta} } }}\overline{n} {\left( {\upzeta} \right)}} \right)}ds,\quad z \notin {\upgamma} ,} }$$
admits a continuous extension to γ and a version of the Sokhotski–Plemelj formulas holds.


Continuous Limit Continuous Density Beltrami Equation Regular Curve Lipschitz Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Begehr, Complex Analytic Methods for Partial Differential Equations. An Introductory Text, World Scientific, River Edge, NJ (1994).zbMATHGoogle Scholar
  2. 2.
    B. Bojarski, “Old and new on Beltrami equations,” Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations: Proc. of the ICTP (February, 8–19, 1988, Trieste, Italy), (1988), pp. 173–187.Google Scholar
  3. 3.
    T. Iwaniec and G. Martin, “What’s new for the Beltrami equation?,” Geometric Analysis and Applications: Proc. Centre Math. Appl. Austral. Nat. Univ. (Canberra, 2000), 39, 132–148 (2001).MathSciNetGoogle Scholar
  4. 4.
    A. Tungatarov, “Properties of an integral operator in classes of summable functions,” Izv. Akad. Nauk Kazakh. SSR, Ser. Fiz.-Mat., No. 5, 58–62 (1985).Google Scholar
  5. 5.
    F. D. Gakhov, Boundary-Value Problems [in Russian], Nauka, Moscow (1977).zbMATHGoogle Scholar
  6. 6.
    N. I. Muskhelishvili, Singular Integral Equations. Boundary-Value Problems in the Theory of Functions and Their Applications in Mathematical Physics, 3 rd edition [in Russian], Nauka, Moscow (1968).Google Scholar
  7. 7.
    N. A. Davydov, “On the continuity of an integral of Cauchy type in a closed region,” Dokl. Akad. Nauk SSSR, 64, 759–762 (1949).zbMATHMathSciNetGoogle Scholar
  8. 8.
    V. V. Salaev and A. O. Tokov, “Necessary and sufficient conditions for the continuity of the Cauchy integral in a closed domain,” Dokl. Akad. Nauk Azerb. SSR., 39, No. 12, 7–11 (1983).MathSciNetGoogle Scholar
  9. 9.
    R. Abreu Blaya, J. Bory Reyes, O. Gerus, and M. Shapiro, “The Clifford–Cauchy transform with a continuous density: N. Davydov theorem,” Math. Meth. Appl. Sci., 28, No. 7, 811–825 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Abreu Blaya, J. Bory Reyes, and M. Shapiro, “On the Laplacian vector fields theory in domains with rectifiable boundary,” Math. Meth. Appl. Sci., 29, No. 15, 1861–1881 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    O. Gerus and M. Shapiro, “On a Cauchy-type integral related to the Helmholtz operator in the plane,” Bol. Soc. Mat. Mexicana, 3, No. 10, 63–82 (2004).MathSciNetGoogle Scholar
  12. 12.
    B. A. Kats, “A generalization of a theorem of N. A. Davydov,” Dokl. Ros. Akad. Nauk, 374, No. 4, 443–444 (2000); English translation: Dokl. Math., 62, 220–221 (2000).MathSciNetGoogle Scholar
  13. 13.
    B. A. Kats, “On a generalization of a theorem of N. A. Davydov,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 1, 39–44 (2002); English translation: Russian Math., 46, No. 1, 37–42 (2002).Google Scholar
  14. 14.
    R. Abreu Blaya, D. Peña Peña, and J. Bory Reyes, “On the jump problem for β-analytic functions,” Compl. Var. Theory Ellipt. Equat., 51, No. 8–11, 763–775 (2006).zbMATHCrossRefGoogle Scholar
  15. 15.
    J. Bory Reyes and D. Peña Peña, “Some higher order Cauchy–Pompeiu integral representations,” Int. Trans. Spec. Funct., 16, No. 8, 615–624 (2005).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Universidad de HolguínHolguínCuba
  2. 2.Universidad de OrienteSantiago de CubaCuba

Personalised recommendations