Skip to main content
SpringerLink
Log in

An analog of Young’s inequality for convolutions of functions for general Morrey-type spaces

Proceedings of the Steklov Institute of Mathematics volume 293, pages 107–126 (2016)Cite this article

Abstract

An analog of the classical Young’s inequality for convolutions of functions is proved in the case of general global Morrey-type spaces. The form of this analog is different from Young’s inequality for convolutions in the case of Lebesgue spaces. A separate analysis is performed for the case of periodic functions.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems, 2nd ed. (Nauka, Moscow, 1996) [in Russian]; Engl. transl. of the 1st ed.: Integral Representations of Functions and Embedding Theorems (J. Wiley & Sons, New York, 1978, 1979).

    MATH  Google Scholar 

  2. V. I. Burenkov, “Sharp estimates for integrals over small intervals for functions possessing some smoothness,” in Progress in Analysis: Proc. 3rd Int. ISAAC Congr., Berlin, 2001 (World Sci., River Edge, NJ, 2003), Vol. 1, pp. 45–56.

    Google Scholar 

  3. V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I, II,” Eurasian Math. J. 3 (3), 8–27 (2012); 4 (1), 21–45 (2013).

    MathSciNet  MATH  Google Scholar 

  4. V. I. Burenkov and W. D. Evans, “The weight Hardy inequality for differences and the complete continuity of the embedding of Sobolev spaces for domains with arbitrary strong degeneracy,” Dokl. Akad. Nauk 355 (5), 583–585 (1997) [Dokl. Math. 56 (1), 565–567 (1997)].

    MathSciNet  MATH  Google Scholar 

  5. V. I. Burenkov and W. D. Evans, “Weighted Hardy-type inequalities for differences and the extension problem for spaces with generalized smoothness,” J. London Math. Soc., Ser. 2, 57 (1), 209–230 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. I. Burenkov, W. D. Evans, and M. L. Goldman, “On weighted Hardy and Poincaré-type inequalities for differences,” J. Inequal. Appl. 1 (1), 1–10 (1997).

    MathSciNet  MATH  Google Scholar 

  7. V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, and R. Ch. Mustafayev, “Boundedness of the fractional maximal operator in local Morrey-type spaces,” Complex Var. Elliptic Eqns. 55 (8–10), 739–758 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  8. V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, and R. Ch. Mustafayev, “Boundedness of the Riesz potential in local Morrey-type spaces,” Potential Anal. 35 (1), 67–87 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. I. Burenkov and M. L. Goldman, “Exact analogues of the Hardy inequality for differences in the case of related weights,” Dokl. Akad. Nauk 366 (2), 155–157 (1999) [Dokl. Math. 59 (3), 372–374 (1999)].

    MathSciNet  MATH  Google Scholar 

  10. V. I. Burenkov and M. L. Goldman, “Hardy-type inequalities for moduli of continuity,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 227, 92–108 (1999) [Proc. Steklov Inst. Math. 227, 87–103 (1999)].

    MathSciNet  MATH  Google Scholar 

  11. V. I. Burenkov and M. L. Goldman, “Necessary and sufficient conditions for the boundedness of the maximal operator from Lebesgue spaces to Morrey-type spaces,” Math. Inequal. Appl. 17 (2), 401–418 (2014).

    MathSciNet  MATH  Google Scholar 

  12. V. I. Burenkov and G. V. Guliev, “Necessary and sufficient conditions for the boundedness of the maximal operator in local Morrey-type spaces,” Dokl. Akad. Nauk 391 (5), 591–594 (2003) [Dokl. Math. 68 (1), 107–110 (2003)].

    MathSciNet  MATH  Google Scholar 

  13. V. I. Burenkov and H. V. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces,” Stud. Math. 163 (2), 157–176 (2004).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. V. I. Burenkov and V. S. Guliyev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces,” Potential Anal. 30 (3), 211–249 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. I. Burenkov, V. S. Guliev, and G. V. Guliev, “Necessary and sufficient conditions for the boundedness of the fractional maximal operator in local Morrey-type spaces,” Dokl. Akad. Nauk 409 (4), 443–447 (2006) [Dokl. Math. 74 (1), 540–544 (2006)].

    MathSciNet  MATH  Google Scholar 

  17. V. I. Burenkov, V. S. Guliev, and G. V. Guliev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces,” Dokl. Akad. Nauk 412 (5), 585–589 (2007) [Dokl. Math. 75 (1), 103–107 (2007)].

    MathSciNet  MATH  Google Scholar 

  18. V. I. Burenkov, V. S. Guliyev, A. Serbetci, and T. V. Tararykova, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces,” Eurasian Math. J. 1 (1), 32–53 (2010).

    MathSciNet  MATH  Google Scholar 

  19. V. I. Burenkov, V. S. Guliev, T. V. Tararykova, and A. Serbetci, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces,” Dokl. Akad. Nauk 422 (1), 11–14 (2008) [Dokl. Math. 78 (2), 651–654 (2008)].

    MathSciNet  MATH  Google Scholar 

  20. V. I. Burenkov, P. Jain, and T. V. Tararykova, “On boundedness of the Hardy operator in Morrey-type spaces,” Eurasian Math. J. 2 (1), 52–80 (2011).

    MathSciNet  MATH  Google Scholar 

  21. V. I. Burenkov, E. D. Nursultanov, and D. K. Chigambayeva, “Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 284, 105–137 (2014) [Proc. Steklov Inst. Math. 284, 97–128 (2014)].

    MathSciNet  MATH  Google Scholar 

  22. V. I. Burenkov and R. Oinarov, “Necessary and sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesgue space to a Morrey-type space,” Math. Inequal. Appl. 16 (1), 1–19 (2013).

    MathSciNet  MATH  Google Scholar 

  23. V. I. Burenkov and Y. Sawano, “Necessary and sufficient conditions for the boundedness of classical operators of real analysis in general Morrey-type spaces,” in Proc. Symp. on Real Analysis, Ibaraki Univ., 2012 (Ibaraki Univ., Mito, 2013), pp. 81–90.

    Google Scholar 

  24. V. S. Guliyev, “Generalized weighted Morrey spaces and higher order commutators of sublinear operators,” Eurasian Math. J. 3 (3), 33–61 (2012).

    MathSciNet  MATH  Google Scholar 

  25. K. Kurata, S. Nishigaki, and S. Sugano, “Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators,” Proc. Am. Math. Soc. 128 (4), 1125–1134 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Kuttner, “Some theorems on fractional derivatives,” Proc. London Math. Soc., Ser. 3, 3, 480–497 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  27. Yu. V. Kuznetsov, “On the pasting of functions from the spaces W p,θ r,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 140, 191–200 (1976) [Proc. Steklov Inst. Math. 140, 209–220 (1976)].

    Google Scholar 

  28. P. G. Lemarié-Rieusset, “The role of Morrey spaces in the study of Navier–Stokes and Euler equations,” Eurasian Math. J. 3 (3), 62–93 (2012).

    MathSciNet  MATH  Google Scholar 

  29. T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces,” in Harmonic Analysis: Proc. Conf. Sendai (Japan), 1990, ICM-90 Satell. Conf. Proc., Ed. by S. Igari (Springer, Tokyo, 1991), pp. 183–189.

    Google Scholar 

  30. E. Nakai, “Recent topics of fractional integrals,” Sugaku Expo. 20 (2), 215–235 (2007).

    MathSciNet  MATH  Google Scholar 

  31. S. M. Nikol’skii, “On a property of the classes H p (r),” Ann. Univ. Sci. Budapest. Eötvös, Sect. Math. 3–4, 205–216 (1960/1961).

    MathSciNet  Google Scholar 

  32. S. M. Nikol’skii, “On imbedding, continuation and approximation theorems for differentiable functions of several variables,” Usp. Mat. Nauk 16 (5), 63–114 (1961) [Russ. Math. Surv. 16 (5), 55–104 (1961)].

    MathSciNet  Google Scholar 

  33. S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems, 2nd ed. (Nauka, Moscow, 1977) [in Russian]; Engl. transl. of the 1st ed.: Approximation of Functions of Several Variables and Imbedding Theorems (Springer, Berlin, 1975).

    Google Scholar 

  34. H. Rafeiro, N. Samko, and S. Samko, “Morrey–Campanato spaces: An overview,” in Operator Theory, Pseudodifferential Equations, and Mathematical Physics (Birkhäuser, Basel, 2013), Oper. Theory Adv. Appl. 228, pp. 293–323.

    Google Scholar 

  35. M. A. Ragusa, “Operators in Morrey type spaces and applications,” Eurasian Math. J. 3 (3), 94–109 (2012).

    MathSciNet  MATH  Google Scholar 

  36. W. Sickel, “Smoothness spaces related to Morrey spaces—a survey. I, II,” Eurasian Math. J. 3 (3), 110–149 (2012); 4 (1), 82–124 (2013).

    MathSciNet  MATH  Google Scholar 

  37. T. V. Tararykova, “Comments on definitions of general local and global Morrey-type spaces,” Eurasian Math. J. 4 (1), 125–134 (2013).

    MathSciNet  MATH  Google Scholar 

  38. H. Triebel, Theory of Function Spaces (Birkhäuser, Basel, 1983).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    V. I. Burenkov

  2. School of Mathematics, Cardiff University, Senghennydd Road, CF24 4AG, Cardiff, Wales, UK

    T. V. Tararykova

Authors
  1. V. I. Burenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. V. Tararykova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Burenkov.

Additional information

Original Russian Text © V.I. Burenkov, T.V. Tararykova, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 293, pp. 113–132.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burenkov, V.I., Tararykova, T.V. An analog of Young’s inequality for convolutions of functions for general Morrey-type spaces. Proc. Steklov Inst. Math. 293, 107–126 (2016). https://doi.org/10.1134/S0081543816040088

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543816040088

search

Navigation