Advertisement

Bounded Operators on Mixed Norm Lebesgue Spaces

Article

Abstract

We study two classes of bounded operators on mixed norm Lebesgue spaces, namely composition operators and product operators. A complete description of bounded composition operators on mixed norm Lebesgue spaces are given (in the case when the inducing mapping preserve the priority of variables). For a certain class of integral operators, we provide sufficient conditions for boundedness. We conclude by applying the developed technique to the investigation of Hardy–Steklov type operators.

Keywords

Mixed norm Lebesgue spaces Composition operator Hardy operator 

Mathematics Subject Classification

Primary 47B33 Secondary 47G10 

Notes

Acknowledgements

Authors gratefully acknowledge the guidance of Prof. Vodopyanov and wish to thank Prof. Jain for bringing the theory of mixed norm spaces to their attention. We are also grateful to the referees for their constructive input.

References

  1. 1.
    Kolyada, V.I.: Mixed norms and Sobolev type inequalities. Banach Center Publ. 72, 141 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Krylov, N.V.: Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. J. Funct. Anal. 250(2), 521–558 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berhanu, S., Cordaro, P.D., Hounie, J.: An Introduction to Involutive Structures. Cambridge University Press, Cambridge (2008)CrossRefMATHGoogle Scholar
  4. 4.
    Antonić, N., Ivec, I.: On the Hörmander–Mihlin theorem for mixed-norm Lebesgue spaces. J. Math. Anal. Appl. 433(1), 176–199 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bandaliyev, R.A.: Connection of two nonlinear differential equations with a two-dimensional Hardy operator in weighted Lebesgue spaces with mixed norms. Electron J. Differ. Equ. 316, 10 (2016)MathSciNetMATHGoogle Scholar
  6. 6.
    Li, S., Stević, S.: Integral type operators from mixed-norm spaces to \(\alpha \)-Bloch spaces. Integral Transf. Spec. Funct. 18(7–8), 485–493 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bandaliev, R.A.: On an inequality in a Lebesgue space with mixed norm and variable summability exponent. Mat. Zametki. 84(3), 323–333 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Delgado, J., Ruzhansky, M., Wang, B.: Approximation property and nuclearity on mixed-norm \(L^p\), modulation and Wiener amalgam spaces. J. Lond. Math. Soc. 94(2), 391–408 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Stević, S.: Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Sib. Mat. Zh. 50(4), 915–927 (2009)MathSciNetMATHGoogle Scholar
  10. 10.
    Stević, S.: Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211(1), 222–233 (2009)MathSciNetMATHGoogle Scholar
  11. 11.
    Zhang, F., Liu, Y.: Products of multiplication, composition and differentiation operators from mixed-norm spaces to weighted-type spaces. Taiwan. J. Math. 18(6), 1927–1940 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jain, P.K., Jain, P., Gupta, B.: On certain weighted integral inequalities with mixed norm. Ital. J. Pure Appl. Math. 18, 23–32 (2005)MathSciNetMATHGoogle Scholar
  13. 13.
    Fiorenza, A., Gupta, B., Jain, P.: Compactness of integral operators in Lebesgue spaces with mixed norm. Math. Inequal. Appl. 11(2), 335–348 (2008)MathSciNetMATHGoogle Scholar
  14. 14.
    Appell, J., Kufner, A.: On the two-dimensional Hardy operator in Lebesgue spaces with mixed norms. Analysis 15(1), 91–98 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grey, W., Sinnamon, G.: Product operators on mixed norm spaces. Linear Nonlinear Anal. 2(2), 189–197 (2016)MathSciNetMATHGoogle Scholar
  16. 16.
    Besov, O.: Estimates of derivatives in a mixed \(L^p\)-norm on a domain and an extension of functions. Math. Notes. 7, 89–94 (1970)CrossRefMATHGoogle Scholar
  17. 17.
    Blozinski, A.: Multivariate rearrangements and Banach function spaces with mixed norms. Trans. Am. Math. Soc. 263, 149–167 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Evseev, N.A., Menovschikov, A.V: Bounded composition operator on mixed Lebesgue spaces. Math. Notes. 2018 (to appear)Google Scholar
  19. 19.
    Vodop’yanov, S., Ukhlov, A.: Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I. Sib. Adv. Math. 14(4), 78–125 (2004)MATHGoogle Scholar
  20. 20.
    Singh, R., Manhas, J.: Composition operators on function spaces. North-Holland, Amsterdam (1993)MATHGoogle Scholar
  21. 21.
    Hardy, G., Littlewood, J., Pólya, G.: Inequalities, 2nd edn. At the University Press. XII, Cambridge (1952)Google Scholar
  22. 22.
    Stepanov, V., Ushakova, E.: On boundedness of a certain class of Hardy–Steklov type operators in Lebesgue spaces. Banach J. Math. Anal. 4(1), 28–52 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Burenkov, V., Jain, P., Tararykova, T.: On Hardy–Steklov and geometric Steklov operators. Math. Nachr. 280(11), 1244–1256 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations