Geodesic
$N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures
Sbornik. Mathematics, Tome 205 (2014) no. 7, pp. 913-935 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {$N^\pm$-integrals and boundary values of {Cauchy-type} integrals of finite measures},
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     url = {http://geodesic.mathdoc.fr/item/SM_2014_205_7_a0/}
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R. A. Aliyev. $N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures. Sbornik. Mathematics, Tome 205 (2014) no. 7, pp. 913-935. http://geodesic.mathdoc.fr/item/SM_2014_205_7_a0/

[1] V. Smirnoff, “Sur les valeurs limites des fonctions régulières à l'intérieur d'un cercle”, Zhurn. Leningrad. fiz.-matem. ob-va, 2:2 (1928), 22–37 | Zbl

[2] V. P. Khavin, “Granichnye svoistva integralov tipa Koshi i garmonicheski sopryazhennykh funktsii v oblastyakh so spryamlyaemoi granitsei”, Matem. sb., 68(110):4 (1965), 499–517 | MR | Zbl

[3] O. D. Kellogg, “Harmonic functions and Green's integral”, Trans. Amer. Math. Soc., 13:1 (1912), 109–132 | DOI | MR | Zbl

[4] P. L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, 38, Academic Press, New York–London, 1970, xii+258 pp. | MR | Zbl

[5] I. I. Privalov, Granichnye svoistva analiticheskikh funktsii, 2nd ed., GITTL, M.–L., 1950, 336 pp. | MR | Zbl

[6] P. Kusis, Vvedenie v teoriyu prostranstv $H^{p}$, S prilozheniem dokazatelstva Volffa teoremy o korone, Mir, M., 1984, 364 pp. ; P. Koosis, Introduction to $H^p$ spaces, With an appendix on Wolff's proof of the corona theorem, London Math. Soc. Lecture Note Ser., 40, Cambridge Univ. Press, Cambridge–New York, 1980, xv+376 pp. | MR | Zbl | MR | Zbl

[7] P. L. Ulyanov, “Ob $A$-integrale Koshi. I”, UMN, 11:5(71) (1956), 223–229 | MR | Zbl

[8] P. L. Ulyanov, “Ob integralakh tipa Koshi”, Sbornik statei. Posvyaschaetsya akademiku Mikhailu Alekseevichu Lavrentevu k ego shestidesyatiletiyu, Tr. MIAN SSSR, 60, Izd-vo AN SSSR, M., 1961, 262–281 ; P. L. Ul'yanov, “Integrals of Cauchy type”, Amer. Math. Soc. Transl. Ser. 2, 44, 1965, 129–150 | MR | Zbl

[9] I. A. Vinogradova, “Obobschennyi integral i sopryazhennye funktsii”, Matem. sb., 99(141):1 (1976), 84–120 | MR | Zbl

[10] I. A. Vinogradova, “Ob $LG^{*}$-integrale tipa Koshi”, Matem. zametki., 20:5 (1976), 645–653 | MR | Zbl

[11] A. B. Aleksandrov, “Ob $A$-integriruemosti granichnykh znachenii garmonicheskikh funktsii”, Matem. zametki, 30:1 (1981), 59–72 | MR | Zbl

[12] T. S. Salimov, “$A$-integral i granichnye znacheniya analiticheskikh funktsii”, Matem. sb., 136(178):1(5) (1988), 24–40 ; T. S. Salimov, “The $A$-integral and boundary values of analytic functions”, Math. USSR-Sb., 64:1 (1989), 23–39 | MR | Zbl | DOI

[13] R. A. Aliyev, “Existence of angular boundary values and Cauchy–Green formula”, Zhurn. matem. fiz., anal., geom., 7:1 (2011), 3–18 | MR | Zbl

[14] E. C. Titchmarsh, “On conjugate functions”, Proc. London Math. Soc., 29:1 (1929), 49–80 | DOI | MR | Zbl

[15] R. A. Aliev, “O predstavimosti analiticheskikh funktsii po svoim granichnym znacheniyam”, Matem. zametki, 73:1 (2003), 8–21 | DOI | MR | Zbl

[16] S. V. Hruščev, S. A. Vinogradov, “Free interpolation in the space of uniformly convergent Taylor series”, Complex analysis and spectral theory (Leningrad, 1979/1980), Lecture Notes in Math., 864, Springer, Berlin–New York, 1981, 171–213 | DOI | MR | Zbl

[17] M. P. Efimova, “O svoistvakh $Q$-integrala”, Matem. zametki, 90:3 (2011), 340–350 | DOI | MR | Zbl

[18] T. S. Salimov, “K teoreme E. Titchmarsha o sopryazhennoi funktsii”, Proc. A. Razmadze Math. Inst., 102 (1993), 99–114 | MR | Zbl

[19] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, 6-e izd., Nauka, M., 1989, 624 pp. ; A. Kolmogorov, S. Fomine, Éléments de la théorie des fonctions et de l'analyse fonctionnelle, Ellipses, Paris; MIR, M., 1994, 536 pp. | MR | Zbl | Zbl

[20] R. A. Aliev, “On boundary properties of analytical functions”, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 29:4, Mathematics and Mechanics (2009), 3–10 | MR | Zbl

[21] F. D. Gakhov, Kraevye zadachi, 2-e izd., Fizmatgiz, M., 1963, 639 pp. ; F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford–New York–Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.–London, 1966, xix+561 pp. | MR | Zbl | MR | Zbl