Skip to main content
SpringerLink
Log in

On Singular Operators in Vanishing Generalized Variable-Exponent Morrey Spaces and Applications to Bergman-Type Spaces

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We give a proof of the boundedness of the Bergman projection in generalized variable-exponent vanishing Morrey spaces over the unit disc and the upper half-plane. To this end, we prove the boundedness of the Calderón—Zygmund operators on generalized variable-exponent vanishing Morrey spaces. We give the proof of the latter in the general context of real functions on Rn, since it is new in such a setting and is of independent interest. We also study the approximation by mollified dilations and estimate the growth of functions near the boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces (Springer-Verlag, New York, 2000).

    Book  Google Scholar 

  2. P. Duren and A. Schuster, Bergman Spaces, in Mathematical Surveys and Monographs (Amer. Math. Soc, Providence, RI, 2004), Vol. 100.

  3. K. Zhu, Spaces of holomorphic functions in the unit ball, Springer, Graduate texts in Math., 2004

    Google Scholar 

  4. K. Zhu, Operator Theory in Function Spaces, in AMS Mathematical Surveys and Monographs (Amer. Math. Soc, Providence, RI, 2007), Vol. 138.

  5. A. Aleman, S. Pott, and M. C. Reguera, “Sarason conjecture on the Bergman space,” Int. Mathematics. Research. Notices, 14, 4320–4349 (2016).

    MATH  Google Scholar 

  6. D. Bekolle and A. Bonami, “Inegalites a poids pour le noyau de Bergman,” CR Acad. Sci. Paris Ser. AB 286 (18), 775–778 (1978).

    MathSciNet  MATH  Google Scholar 

  7. D. Milutin, “Boundedness of the Bergman projections on L p spaces with radial weights,” Publications de l’Institut Mathematique 86 (100), 5–20 (2009).

    Article  Google Scholar 

  8. V. S. Guliyev and S. G. Samko, “Maximal, potential, and singular operators in the generalized variable-exponent Morrey spaces on unbounded sets,” J. Math. Sci. 193 (2) 228–248 (2013).

    Article  MathSciNet  Google Scholar 

  9. N. Samko, “Maximal, potential, and singular operators in vanishing generalized Morrey spaces,” J. Global Optim. 57(4), 1385–1399 (2013).

    Article  MathSciNet  Google Scholar 

  10. A. Karapetyants and S. Samko, “On Boundedness of Bergman Projection Operators in Banach Spaces of Holomorphic Functions in Half-Plane and Harmonic Functions in Half-Space,” J. Math. Sci. 226 (4), 344–354 (2017).

    Article  MathSciNet  Google Scholar 

  11. A. Karapetyants, H. Rafeiro, and S. Samko, Boundedness of the Bergman Projection and Some Properties of Bergman-Type Spaces, in Complex Analysis and Operator Theory (1–15) (2018); https://doi.org/10.1007/s11785-018-0780-y.

    Google Scholar 

  12. A. Almeida and S. Samko, “Approximation in Morrey spaces,” J. Funct. Anal. 272, 2392–2411 (2017).

    Article  MathSciNet  Google Scholar 

  13. A. Almeida and S. Samko, “Approximation in generalized Morrey spaces,” Georgian Math. J., 2018 (in press).

    Google Scholar 

  14. V. P. Zaharyuta and V. Yudovich, “The general form of a linear functional in H l p Uspekhi Mat. Nauk} 19 (2(116)), 139–142 (1964).

    MathSciNet  Google Scholar 

  15. V. S. Guliyev, J. J. Hasanov, and S. G. Samko, “Boundedness of the maximal, potential and singular operators in the generalized variable-exponent Morrey spaces,” Math. Scand. 107, 285–304 (2010).

    Article  MathSciNet  Google Scholar 

  16. L. Diening and M. Ruzichka, “Calderon-Zygmund operators on generalized Lebesgues spaces L p(.) and problems related to fluid dynamics,” J. Reine Angew. Math. 563, 197–220 (2003).

    MathSciNet  Google Scholar 

  17. G. R. Chacon and H. Rafeiro, “Variable exponent Bergman spaces,” Nonlinear Analysis: Theory, Methods, and Applications 105, 41–49 (2014).

    Article  MathSciNet  Google Scholar 

  18. G. R. Chacon and H. Rafeiro, “Toeplitz operators on variable exponent Bergman spaces,” Mediterr. J. Math. 13 (5), 3525–3536 (2014).

    Article  MathSciNet  Google Scholar 

  19. A. Almeida, J. Hasanov, and S. Samko, “Maximal and potential operators in variable-exponent Morrey spaces,” Georgian Math. J. 15, 195–208 (2008).

    MathSciNet  MATH  Google Scholar 

  20. V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. I: Variable Exponent Lebesgue and Amalgam Spaces, in Operator Theory: Advances and Applications (Birkhauser, Basel, 2016).

    MATH  Google Scholar 

  21. V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. II: Variable Exponent Holder, Morrey—Campanato and Grand Spaces, in Operator Theory: Advances and Applications (Birkhauser, Basel, 2016).

    MATH  Google Scholar 

  22. C. Bennett and R. Sharpley, Interpolation of Operators (Academic Press, Boston, 1988).

    MATH  Google Scholar 

  23. V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev, “Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces,” J. Comput. Appl. Math. 208 (1), 280–301 (2007).

    Article  MathSciNet  Google Scholar 

  24. P. D. Liu, Y. L. Hou, and M. F. Wang, “Weak Orlicz space and its applications to the martingale theory,” Sci. China Math. 53 (4), 905–916 (2010).

    Article  MathSciNet  Google Scholar 

  25. D. Gallardo, “Orlicz spaces for which the Hardy-Littlewood maximal operator is bounded,” Publ. Mat. Barc. 32 (2), 261–266 (1988).

    Article  MathSciNet  Google Scholar 

  26. E. Nakai, “A characterization of pointwise multipliers on the Morrey spaces,” Sci. Math. 3 (3), 445–454 (2000).

    MathSciNet  MATH  Google Scholar 

  27. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, in Lecture Notes in Math. (Springer-Verlag, Heidelberg, (2011), Vol. 2017.

    Google Scholar 

  28. D. Cruz-Uribe and A. Fiorenza Variable Lebesgue Spaces. Foundations and Harmonic Analysis, in Applied and Numerical Harmonic Analysis (Springer, Basel, 2013).

    MATH  Google Scholar 

  29. F. Deringoz, V. S. Guliyev, and S. Samko, “Vanishing generalized Orlicz-Morrey spaces and fractional maximal operator,” Publ. Math. Debrecen 90 (1–2), 125–147, (2017).

    Article  MathSciNet  Google Scholar 

  30. H. Triebel, Local Function Spaces, Heat and Navier-Stokes Equations, in EMS Tracts in Math. (2013), Vol. 20.

  31. L. Pick, A. Kufner, O. John, and S. Fucik, Function Spaces, I, in De Gruyter Series in Nonlinear Analysis and Applications (2013), Vol. 14.

  32. H. Rafeiro, N. Samko, and S. Samko, Morrey-Campanato Spaces: an Overview, in Operator Theory: Advances and Applications (Springer, Basel, 2013), Vol. 228, pp. 293–323.

    MathSciNet  MATH  Google Scholar 

  33. A. Kufner, O. John, and S. Fucik, Function Spaces, Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis (Noordhoff Int. Publ., Leyden, 1977).

    Google Scholar 

  34. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, in Annals of Math. Studies (Princenton University Press, Princenton, NJ, 1977), Vol. 105.

  35. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Math. (Marcel Dekker, 1991), Vol. 146.

  36. V. Kokilashvili and M. M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces (World Scientific, Singapore, 1991).

    Book  Google Scholar 

  37. M. A. Krasnoselskii and Ya. B. Rutitski, Convex Functions and Orlicz Spaces, Translated from the first Russian edition (Noordhoff, Groningen, 1961).

    Google Scholar 

  38. C. Vitanza, “Functions with vanishing Morrey norm and elliptic partial differential equations,” in Proceedings of Methods of Real Analysis and Partial Differential Equations, Capri (Springer, 1990), pp. 147–150.

    Google Scholar 

  39. J. Duoandikoetxea. Fourier Analysis, Graduate Studies (Amer. Math. Soc., 2001), Vol. 29.

Download references

Funding

The work of A. N. Karapetyants and S. G. Samko was supported in part by the Russian Foundation for Basic Research under grant 18-01-00094. The work of A. N. Karapetyants was also supported in part by the Russian Foundation for Basic Research under grant 18-51-05009 Arm-a and by Visiting Fulbright Scholar Program. The research of H. Rafeiro was supported by a Research Start-up Grant of United Arab Emirates University, Al Ain, United Arab Emirates, via grant no. G00002994. The research of S. Samko was supported by the Russian Foundation for Basic Research under grant 19-01-00223.

Author information

Authors and Affiliations

  1. Southern Federal University, Rostov-on-Don, 334006, Russia

    A. N. Karapetyants

  2. State University of New York, Albany, 12222, USA

    A. N. Karapetyants

  3. Department of Mathematical Sciences, College of Sciences, United Arab Emirates University, Al Ain, 15551, UAE

    H. Rafeiro

  4. University of Algarve, Faro, 8005-139, Portugal

    S. G. Samko

Authors
  1. A. N. Karapetyants
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Rafeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. G. Samko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. N. Karapetyants, H. Rafeiro or S. G. Samko.

Additional information

The article was submitted by the authors for the English version of the journal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karapetyants, A.N., Rafeiro, H. & Samko, S.G. On Singular Operators in Vanishing Generalized Variable-Exponent Morrey Spaces and Applications to Bergman-Type Spaces. Math Notes 106, 727–739 (2019). https://doi.org/10.1134/S0001434619110075

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434619110075

Keywords

Search

Navigation