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 Mat. Sb., 2014, Volume 205, Number 7, Pages 3–24 (Mi msb8268)

$N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures

R. A. Aliyev

Baku State University

Abstract: Let $\Gamma$ be a simple closed Lyapunov contour with finite complex measure $\nu$, and let $G^+$ be the bounded and $G^-$ the unbounded domains with boundary $\Gamma$. Using new notions (so-called $N$-integration and $N^+$- and $N^-$-integrals), we prove that the Cauchy-type integrals $F^+(z)$, $z\in G^+$, and $F^-(z)$, $z\in G^-$, of $\nu$ are Cauchy $N^+$- and $N^-$-integrals, respectively. In the proof of the corresponding results, the additivity property and the validity of the change-of-variable formula for the $N^+$- and $N^-$-integrals play an essential role.
Bibliography: 21 titles.

Keywords: finite complex Borel measure, Cauchy-type integral, nontangential boundary values, Cauchy integral, $Q$-integral, $Q'$-integral, $N$-integration.

DOI: https://doi.org/10.4213/sm8268

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English version:
Sbornik: Mathematics, 2014, 205:7, 913–935

Bibliographic databases:

UDC: 517.518.234+517.547.73
MSC: Primary 28A25; Secondary 26A42, 42B25

Citation: R. A. Aliyev, “$N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures”, Mat. Sb., 205:7 (2014), 3–24; Sb. Math., 205:7 (2014), 913–935

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8268
• https://doi.org/10.4213/sm8268
• http://mi.mathnet.ru/eng/msb/v205/i7/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. R. A. Aliev, “On Taylor coefficients of Cauchy-type integrals of finite complex measures”, Complex Var. Elliptic Equ., 60:12 (2015), 1727–1738
2. “On the Properties of Q- and Q `-Integrals of the Function Measurable on the Real Axis”, Proc. Inst. Math. Mech., 41:1 (2015), 56–62
3. R. A. Aliev, “Riesz's equality for the Hilbert transform of the finite complex measures”, Azerb. J. Math., 6:1 (2016), 126–135
4. R. A. Aliev, “On properties of Hilbert transform of finite complex measures”, Complex Anal. Oper. Theory, 10:1 (2016), 171–185
5. R. A. Aliev, “On Laurent coefficients of Cauchy type integrals of finite complex measures”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 42:2 (2016), 292–303
6. R. A. Aliev, “Representability of Cauchy-type integrals of finite complex measures on the real axis in terms of their boundary values”, Complex Var. Elliptic Equ., 62:4 (2017), 536–553
7. R. A. Aliev, Kh. I. Nebiyeva, “The $A$-integral and restricted Ahlfors–Beurling transform”, Integral Transform. Spec. Funct., 29:10 (2018), 820–830
8. Aliev R.A., Amrahova A.F., “Properties of the Discrete Hilbert Transform”, Complex Anal. Oper. Theory, 13:8 (2019), 3883–3897
9. Aliev R.A., Nebiyeva I Kh., “The a-Integral and Restricted Complex Riesz Transform”, Azerbaijan J. Math., 10:1 (2020), 209–221
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