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Atomic Decomposition for Mixed Morrey Spaces

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Abstract

In this paper, we consider some norm estimates for mixed Morrey spaces considered by the first author. Mixed Lebesgue spaces are realized as a special case of mixed Morrey spaces. What is new in this paper is a new norm estimate for mixed Morrey spaces that is applicable to mixed Lebesgue spaces as well. An example shows that the condition on parameters is optimal. As an application, the Olsen inequality adapted to mixed Morrey spaces can be obtained.

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References

  1. Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53(6), 1629–1663 (2004)

    Article  MathSciNet  Google Scholar 

  2. Bagby, R.J.: An extended inequality for the maximal function. Proc. Am. Math. Soc. 48, 419–422 (1975)

    Article  MathSciNet  Google Scholar 

  3. Benedek, A., Panzone, R.: The \(L^{P}\), with mixed norm. Duke Math. J. 28, 301–324 (1961)

    Article  MathSciNet  Google Scholar 

  4. Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. 7, 273–279 (1987)

    MathSciNet  MATH  Google Scholar 

  5. Cruz-Uribe, D., Wang, L.-A.D.: Variable Hardy spaces. Indiana Univ. Math. J. 63(2), 447–493 (2014)

    Article  MathSciNet  Google Scholar 

  6. Duoandikoetxea, J.: Fourier Analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence (2001)

  7. Eridani, U.M.I., Gunawan, A., Utoyo, H.G.: A characterization for fractional integral operators on generalized Morrey spaces. Anal. Theory Appl. 28(3), 263–267 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Eridani, U.M.I., Gunawan, H., Nakai, E., Sawano, Y.: Characterizations for the generalized fractional integral operators on Morrey spaces. Math. Inequal. Appl. (2014). https://doi.org/10.7153/mia-17-56

    Article  MathSciNet  MATH  Google Scholar 

  9. Gogatishvili, A., Mustafayev, R.: New pre-dual space of Morrey space. J. Math. Anal. Appl. 397(2), 678–692 (2013)

    Article  MathSciNet  Google Scholar 

  10. Ho, K.P.: Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces. Anal. Math. 38(3), 173–185 (2012)

    Article  MathSciNet  Google Scholar 

  11. Ho, K.P., Sawano, Y., Yang, D., Yang, S.: Hardy spaces for ball quasi-Banach function spaces. Diss. Math. 525, 1–102 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Iida, T.: Weighted inequalities on Morrey spaces for linear and multilinear fractional integrals with homogeneous kernels. Taiwan. Math. J. 18(1), 147–185 (2014)

    Article  MathSciNet  Google Scholar 

  13. Iida, T., Sawano, Y., Tanaka, H.: Atomic decomposition for Morrey spaces. Z. Anal. Anwend. 33(2), 149–170 (2014)

    Article  MathSciNet  Google Scholar 

  14. Jia, H., Wang, H.: Decomposition of Hardy-Morrey spaces. J. Math. Anal. Appl. 354(1), 99–110 (2009)

    Article  MathSciNet  Google Scholar 

  15. Kalita, E.A.: Dual Morrey spaces. Dokl. Akad. Nauk. 361(4), 447–449 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Long, R.L.: The spaces generated by blocks. Sci. Sinica Ser. A 27(1), 16–26 (1984)

    MathSciNet  MATH  Google Scholar 

  17. Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)

    Article  MathSciNet  Google Scholar 

  18. Nogayama, T.: Mixed Morrey spaces. Positivity 23(4), 961–1000 (2019)

    Article  MathSciNet  Google Scholar 

  19. Nogayama, T.: Boundedness of the commutators of fractional integral operators on mixed Morrey spaces. Integr. Transforms Spec. Funct. 30(10), 790–816 (2019)

    Article  MathSciNet  Google Scholar 

  20. Sawano, Y.: A note on Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces. Acta Math. Sin. (Engl. Ser.) 25, 1223–1242 (2009)

    Article  MathSciNet  Google Scholar 

  21. Sawano, Y.: Theory of Besov spaces. In: Development in Mathematics, vol. 56. Springer, Singapore (2018)

  22. Sawano, Y., Sugano, S., Tanaka, H.: A note on generalized fractional integral operators on generalized Morrey spaces. Bound. Value Probl. Art. ID 835865 (2009)

  23. Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 363(12), 6481–6503 (2011)

    Article  MathSciNet  Google Scholar 

  24. Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin. 21(6), 1535–1544 (2005)

    Article  MathSciNet  Google Scholar 

  25. Stein, E.M.: Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  26. Strömberg, J.O.,Torchinsky, A.: Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag (1989)

  27. Tang, L., Xu, J.: Some properties of Morrey type Besov–Triebel spaces. Math. Nachr. 278(7–8), 904–917 (2005)

    Article  MathSciNet  Google Scholar 

  28. Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer, Berlin (2010)

  29. Zorko, C.: Morrey space. Proc. Am. Math. Soc. 98(4), 586–592 (1986)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Toru Nogayama and Takahiro Ono were supported financially by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (20J10403 and 20J20606). Daniel Salim was supported by P3MI-ITB Program 2017. Yoshihiro Sawano was supported by Grant-in-Aid for Scientific Research (C) (19K03546), the Japan Society for the Promotion of Science and People’s Friendship University of Russia. The authors are thankful to Dr. Naoya Hatano for pointing out the mistake in the proof of Theorem 4.

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Nogayama, T., Ono, T., Salim, D. et al. Atomic Decomposition for Mixed Morrey Spaces. J Geom Anal 31, 9338–9365 (2021). https://doi.org/10.1007/s12220-020-00513-z

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