2017
Том 69
№ 9

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

Abreu Blaya R., Bory Reyes J.

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Abstract

The aim of this paper is to extend a theorem from classical complex analysis proved by Davydov in 1949 to the theory of solutions of a special case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator $\partial_{\overline{z}} - \beta \frac{z}{\overline{z}}\partial_z,\quad 0 \leq \beta < 1$). We prove that if $\gamma$ is a rectifiable Jordan closed curve and $f$ is a continuous complex-valued function on $\gamma$ such that the integral $$\int\limits_{\gamma\setminus\{\zeta \in \gamma:\;|\zeta-t|\leq r \}} \frac{|f(\zeta) - f(t)|}{\left|\zeta - t|t/\zeta|^{\theta} \right|} \left|n(\zeta) - \beta \frac{\zeta}{\overline{\zeta}} \overline{n}(\zeta) \right|ds, \quad \theta = \frac{2\beta}{1-\beta},$$ converges uniformly on $\gamma$ as $r \rightarrow 0$, where $n(\zeta)$ is the exterior unit normal vector on $\gamma$ at a point $n(\zeta)$ and $ds$ is the arc length differential, then the $\beta$-Cauchy type integral $$\frac1{2(1 - \beta)\pi}\int\limits_{\gamma}\frac{f(\zeta)}{\zeta - z|z/\zeta|^ {\theta}} \left(n(\zeta) - \beta \frac{\zeta}{\overline{\zeta}}\overline{n}(\zeta) \right)ds,\quad z \in \gamma,$$ admits a continuous extension to $\gamma$ and a version of the Sokhotski - Plemelj formulae holds. Метою цієї статті є узагальнення теореми із класичного комплексного аналізу, що була доведена Давидовим у 1949 р., для теорії розв'язків окремого випадку рівняння Вельтрамі у $z$-комплексній площині (тобто нульових розв'язків диференціального оператора $\partial_{\overline{z}} - \beta \frac{z}{\overline{z}}\partial_z,\quad 0 \leq \beta < 1$).

English version (Springer): Ukrainian Mathematical Journal 60 (2008), no. 11, pp 1683-1690.

Citation Example: Abreu Blaya R., Bory Reyes J. A property of the β-Cauchy-type integral with continuous density // Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1443–1448.

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