Ukrainian Mathematical Journal

, Volume 60, Issue 11, pp 1683–1690

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

• R. Abreu Blaya
• J. Bory Reyes
Article

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.

## Keywords

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.

## References

1. 1.
H. Begehr, Complex Analytic Methods for Partial Differential Equations. An Introductory Text, World Scientific, River Edge, NJ (1994).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).