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A boundedness criterion for Toeplitz operators in weighted Sobolev spaces of holomorphic functions on the polydisk

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Abstract

We obtain a full description of the summable functions h on the torus which admit pluriharmonic continuation to the unit polydisk for which the Toeplitz operator with symbol h is a bounded operator in the weighted Sobolev spaces of holomorphic functions.

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References

  1. Carleson L., “On the zeros of functions with bounded Dirichlet integral,” Math. Z., Bd 3, 289–295 (1952).

    Article  MathSciNet  Google Scholar 

  2. Korenblyum B. I., “An extremal property of outer functions,” Math. Notes, 10, No. 1, 456–458 (1971).

    Article  Google Scholar 

  3. Korenblyum B. I., “Functions holomorphic in a disk and smooth in its closure,” Soviet Math., Dokl., 12, No. 1, 1312–1315 (1971).

    MathSciNet  MATH  Google Scholar 

  4. Khavin V. P., “On factorization of analytic functions that are smooth up to a boundary,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 22, 202–205 (1971).

    MATH  Google Scholar 

  5. Shamoyan F. A., “Division by an inner function in some spaces of functions analytic in the disk,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 22, 206–208 (1971).

    MATH  Google Scholar 

  6. Vinogradov S. A. and Shirokov N. A., “On the factorization of analytic functions with the derivative in H p,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 22, 8–27 (1971).

    MATH  Google Scholar 

  7. Shirokov N. A., Analytic Functions Smooth up to the Boundary, Springer-Verlag, Berlin (1988) (Lecture Notes in Math.; V. 1312).

    MATH  Google Scholar 

  8. Dyakonov K. M., “Division and multiplication by Inner Functions And Embedding theorems for spaces of analytic functions,” Amer. J. Math., 115, No. 4, 881–902 (1983).

    Article  MathSciNet  Google Scholar 

  9. Shamoyan F. A. and Shubabko E. N., Introduction in the Theory of Weighted L p-Classes of Meromorphic Functions [in Russian], Bryansk Univ., Bryansk (2010).

    Google Scholar 

  10. Shamoyan F. A., “On boundedness of one class of operators connected with divisibility of analytic functions,” Izv. Akad. Nauk Armenii Mat., 8,No. 6, 474–494 (1973).

    Google Scholar 

  11. Shamoyan F. A., “On a class of operators connected with the factorization of analytic functions,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 39, 200–206 (1974).

    MATH  Google Scholar 

  12. Songxiao L. and Stevo S., “Volterra type operators from Zygmund space into Bloch spaces,” J. Concr. Appl. Math., 6,No. 2, 199–207 (2008).

    MathSciNet  MATH  Google Scholar 

  13. Jonson S., Peetre J., and Semmes A., “On action of Hankel and Toeplitz operators on some function spaces,” Duke Math. J., 51, No. 4, 937–958 (1984).

    Article  MathSciNet  Google Scholar 

  14. Zhu K., “Multipliers of BMO in the Bergman metric with applications to Toeplitz operators,” J. Funct. Anal., 87, No. 1, 31–50 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  15. Shamoyan F. A., “Toeplitz operators and division by an inner function in some spaces of analytic functions,” Dokl. Akad. Nauk Armenii, 76,No. 3, 215–219 (1983).

    MathSciNet  Google Scholar 

  16. Jöricke B., “The multidimensional analog of the Privalov theorem,” Math. Nachr., Bd 107, 221–233 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  17. Rudin W., Function Theory in Polydisks, W. A. Benjamin, New York and Amsterdam (1969).

    Google Scholar 

  18. Shamoyan F. A., “Applications of Dzrbasjan integral representations to some problems in analysis,” Soviet Math., Dokl., 24, 563–567 (1981).

    MATH  Google Scholar 

  19. Shamoyan F. A., Factorization, Integral Representations, and Ideals of Analytic Functions [in Russian], Diss. Dokt. Fiz.-Mat. Nauk, Steklov Institute, Leningrad (1983).

    Google Scholar 

  20. Shamoyan F. A., “Diagonal mappings and problems of representation in anisotropic spaces of holomorphic functions in the polydisk,” Siberian Math. J., 31,No. 2, 350–365 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  21. Shamoyan F. A. and Arutyunyan A. V., “Toeplitz operators in anisotropic spaces of holomorphic functions in a polydisk,” Dokl. Akad. Nauk Armenii, 91, No. 4, 147–151 (1990).

    MathSciNet  Google Scholar 

  22. Antonenkova O. E. and Shamoyan F. A., “The Cauchy transform of continuous linear functionals and projections on the weighted spaces of analytic functions,” Siberian Math. J., 46, No. 6, 969–994 (2005).

    Article  MathSciNet  Google Scholar 

  23. Tkachenko N. M. and Shamoyan F. A., “The Hardy-Littlewood theorem and the operator of harmonic conjugate in some classes of simply connected domains with rectifiable boundary,” J. Math. Phys., Anal., Geom., 5, No. 2, 192–210 (2009).

    MathSciNet  MATH  Google Scholar 

  24. Shamoyan F. A., “Embedding theorems and a characterization of traces in the spaces H p(U n), 0 < p < +∞,” Math. USSR-Sb., 35,No. 5, 709–725 (1979).

    Article  MATH  Google Scholar 

  25. Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970).

    MATH  Google Scholar 

  26. Triebel H., Theory of Function Spaces, Birkhäuser, Basel (2010).

    Google Scholar 

  27. Kahane J. P., “Best approximation in L 1(T),” Bull. Amer. Math. Soc., 80, No. 5, 788–804 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  28. Nikolskii N. K., Operators, Functions and Systems: An Easy Reading, Amer. Math. Soc., Providence (2002). (Math. Surveys and Monographs; 92.)

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Authors and Affiliations

  1. Bryansk State University, Bryansk, Russia

    F. A. Shamoyan

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  1. F. A. Shamoyan
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Correspondence to F. A. Shamoyan.

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Original Russian Text Copyright © 2012 Shamoyan F. A.

The author was supported by the Russian Foundation for Basic Research (Grant 09-01-97517).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 3, pp. 691–711, May–June, 2012.

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Shamoyan, F.A. A boundedness criterion for Toeplitz operators in weighted Sobolev spaces of holomorphic functions on the polydisk. Sib Math J 53, 554–572 (2012). https://doi.org/10.1134/S003744661202036X

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  • DOI: https://doi.org/10.1134/S003744661202036X

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