The Riemann boundary value problem on non-rectifiable curve and the Cauchy transform
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 152 (2010) no. 1, pp. 164-173
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In the present paper we obtain a representation for solutions of the Riemann boundary value problem on non-rectifiable closed Jordan curves in terms of the Cauchy transforms of certain distributions.
Mots-clés : non-rectifiable curve, Cauchy transform.
Keywords: Riemann boundary value problem
@article{UZKU_2010_152_1_a15,
     author = {B. A. Kats},
     title = {The {Riemann} boundary value problem on non-rectifiable curve and the {Cauchy} transform},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {164--173},
     year = {2010},
     volume = {152},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2010_152_1_a15/}
}
TY  - JOUR
AU  - B. A. Kats
TI  - The Riemann boundary value problem on non-rectifiable curve and the Cauchy transform
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2010
SP  - 164
EP  - 173
VL  - 152
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UZKU_2010_152_1_a15/
LA  - ru
ID  - UZKU_2010_152_1_a15
ER  - 
%0 Journal Article
%A B. A. Kats
%T The Riemann boundary value problem on non-rectifiable curve and the Cauchy transform
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2010
%P 164-173
%V 152
%N 1
%U http://geodesic.mathdoc.fr/item/UZKU_2010_152_1_a15/
%G ru
%F UZKU_2010_152_1_a15
B. A. Kats. The Riemann boundary value problem on non-rectifiable curve and the Cauchy transform. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 152 (2010) no. 1, pp. 164-173. http://geodesic.mathdoc.fr/item/UZKU_2010_152_1_a15/

[1] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977, 640 pp. | MR | Zbl

[2] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1962, 600 pp. | MR

[3] Kats B. A., “Zadacha Rimana na zamknutoi zhordanovoi krivoi”, Izv. vuzov. Matem., 1983, no. 4, 68–80 | MR | Zbl

[4] Kolmogorov A. N., Tikhomirov V. M., “$\varepsilon$-entropiya i $\varepsilon$-emkost mnozhestv v funktsionalnykh prostranstvakh”, Usp. matem. nauk, 14:2 (1959), 3–86 | MR | Zbl

[5] Feder E., Fraktaly, Mir, M., 1991, 254 pp. | MR

[6] Mattila P., Melnikov M. S., “Existence and weak type inequalities for Cauchy integrals of general measure on rectifiable curves and sets”, Proc. Am. Math. Soc., 120 (1994), 143–149 | DOI | MR | Zbl

[7] Tolsa X., “Bilipschitz maps, analytic capacity and the Cauchy integral”, Ann. of Math., 162:2 (2005), 1243–1304 | DOI | MR | Zbl

[8] Verdera J., “A weak type inequality for Cauchy transform of finite measure”, Publ. Mat., 36 (1992), 1029–1034 | DOI | MR | Zbl

[9] Hörmander L., The Analysis of Linear Partial Differential Operators, v. I, Distribution theory and Fourier Analysis, Springer Verlag, 1983, 464 pp. | MR

[10] Kats B. A., “Zadacha o skachke i integral po nespryamlyaemoi krivoi”, Izv. vuzov. Matem., 1987, no. 5, 49–57 | MR | Zbl

[11] Kats B. A., “The Cauchy integral over non-rectifiable paths”, Contemp. Math., 455 (2008), 183–196 | DOI | MR | Zbl

[12] Harrison J., Norton A., “Geometric integration on fractal curves in the plane”, Indiana Univ. Math. J., 40:2 (1991), 567–594 | DOI | MR | Zbl

[13] Harrison J., Lectures on chainlet geometry – new topological methods in geometric measure theory, 24 May 2005, 153 pp., arXiv: math-ph/0505063

[14] Kats B. A., “On solvability of the jump problem”, J. Math. Anal. Appl., 356:2 (2009), 577–581 | DOI | MR | Zbl

[15] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978, 400 pp. | MR

[16] Vekua I. N., Obobschennye analiticheskie funktsii, Nauka, M., 1988, 512 pp. | MR | Zbl

[17] Dolzhenko E. P., “O “stiranii” osobennostei analiticheskikh funktsii”, Usp. matem. nauk, 18:4 (1963), 135–142 | MR | Zbl