Abstract:
Let ΓΓ be a simple closed Lyapunov contour with finite complex measure νν, and let G+G+ be the bounded and G−G− the unbounded domains with boundary ΓΓ. Using new notions (so-called NN-integration and N+N+- and N−N−-integrals), we prove that the Cauchy-type integrals F+(z)F+(z), z∈G+z∈G+, and F−(z)F−(z), z∈G−z∈G−, of νν are Cauchy N+N+- and N−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+N+- and N−N−-integrals play an essential role.
Bibliography: 21 titles.
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\by R.~A.~Aliyev
\paper $N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures
\jour Sb. Math.
\yr 2014
\vol 205
\issue 7
\pages 913--935
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https://doi.org/10.1070/SM2014v205n07ABEH004403
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This publication is cited in the following 10 articles:
R. A. Aliev, Kh. I. Nebiyeva, “The a-integral and restricted complex riesz transform”, Azerbaijan J. Math., 10:1 (2020), 209–221
Rashid ALİEV, Khanim NEBİYEVA, “The A-Integral and Restricted Riesz Transform”, Constructive Mathematical Analysis, 3:3 (2020), 104
R. A. Aliev, A. F. Amrahova, “Properties of the discrete Hilbert transform”, Complex Anal. Oper. Theory, 13:8 (2019), 3883–3897
R. A. Aliev, Kh. I. Nebiyeva, “The $A$-integral and restricted Ahlfors–Beurling transform”, Integral Transform. Spec. Funct., 29:10 (2018), 820–830
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
R. A. Aliev, “Riesz's equality for the Hilbert transform of the finite complex measures”, Azerb. J. Math., 6:1 (2016), 126–135
R. A. Aliev, “On properties of Hilbert transform of finite complex measures”, Complex Anal. Oper. Theory, 10:1 (2016), 171–185
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
R. A. Aliev, “On Taylor coefficients of Cauchy-type integrals of finite complex measures”, Complex Var. Elliptic Equ., 60:12 (2015), 1727–1738
“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
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