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Boundary values of Cauchy type integrals


Author: V. V. Kapustin
Translated by: the author
Original publication: Algebra i Analiz, tom 16 (2004), nomer 4.
Journal: St. Petersburg Math. J. 16 (2005), 691-702
MSC (2000): Primary 30E20, 47B47
DOI: https://doi.org/10.1090/S1061-0022-05-00873-3
Published electronically: June 23, 2005
MathSciNet review: 2090853
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Abstract: Results by A. G. Poltoratskii and A. B. Aleksandrov about nontangential boundary values of pseudocontinuable $H^2$-functions on sets of zero Lebesgue measure are used for the study of operators on $L^2$-spaces on the unit circle. For an arbitrary bounded operator $X$ acting from one such $L^2$-space to another and having the property that the commutator of it with multiplication by the independent variable is a rank one operator, it is shown that $X$ can be represented as a sum of multiplication by a function and a Cauchy transformation in the sense of angular boundary values.


References [Enhancements On Off] (What's this?)

  • 1. A. G. Poltoratskii, The boundary behavior of pseudocontinuable functions, Algebra i Analiz 5 (1993), no. 2, 189-210; English transl., St. Petersburg Math. J. 5 (1994), no. 2, 389-406. MR 1223178 (94k:30090)
  • 2. A. B. Aleksandrov, On the existence of nontangential boundary values of pseudocontinuable functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), 5-17; English transl., J. Math. Sci. 87 (1997), no. 5, 3781-3787. MR 1359992 (97a:30046)
  • 3. N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2. Model operators and systems, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, RI, 2002. MR 1892647 (2003i:47001b)
  • 4. A. Volberg, Calderón-Zygmund capacities and operators on nonhomogeneous spaces, CBMS Regional Conf. Ser. in Math., vol. 100, Amer. Math. Soc., Providence, RI, 2003. MR 2019058
  • 5. F. Nazarov, S. Treil, and A. Volberg, Two-weight Hilbert transform, Preprint, 2003.
  • 6. F. Nazarov and A. Volberg, The Bellman function, the two-weight Hilbert transform, and embeddings of the model spaces $K_\theta$, J. Anal. Math. 87 (2002), 385-414. MR 1945290 (2003j:30081)
  • 7. P. Wojtaszczyk, Banach spaces for analysts, Cambridge Stud. Adv. Math., vol. 25, Cambridge Univ. Press, Cambridge, 1991. MR 1144277 (93d:46001)
  • 8. D. R. Yafaev, Mathematical scattering theory. General theory, S.-Peterburg. Univ., St. Petersburg, 1994; English transl., Transl. Math. Monogr., vol. 105, Amer. Math. Soc., Providence, RI, 1992. MR 1784870 (2001e:47015); MR 1180965 (94f:47012)
  • 9. D. Clark, One-dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972), 169-191. MR 0301534 (46:692)

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Additional Information

V. V. Kapustin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: kapustin@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-05-00873-3
Keywords: Cauchy type integral, angular boundary values, intertwining relations
Received by editor(s): January 20, 2004
Published electronically: June 23, 2005
Additional Notes: Partially supported by RFBR (grant no. 02–01–00264), and by the SS grant no. 2266.2003.1.
Article copyright: © Copyright 2005 American Mathematical Society

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