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The Fractional Operators on Weighted Morrey Spaces

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Abstract

A necessary condition and a sufficient condition for one weight norm inequalities on Morrey spaces to hold are given for the fractional maximal operator and the fractional integral operator. The difference between the behavior of the fractional maximal operator and one of the fractional integral operator which originates from the structure of Morrey spaces will be clarified. The necessary and sufficient conditions are also verified for the power weights.

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References

  1. Adams, D.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, D., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50(2), 201–230 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Math. Appl. 7, 273–279 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Coifman, R.R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  5. Garcia-Cuerva, J.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies. Elsevier, Amsterdam (1985)

    MATH  Google Scholar 

  6. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag, Berlin (1983)

    Book  MATH  Google Scholar 

  7. Haroske, D.D., Skrzypczak, L.: Embeddings of weighted Morrey spaces. Math. Nachr. 290, 1066–1086 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hytönen, T.: The sharp weighted bound for general Calderòn-Zygmund operators. Ann. of Math. 175(2), 1473–1506 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hytönen, T., Perez, C.: Sharp weighted bounds involving \(A_{\infty }\). Anal. PDE 6, 777–818 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_{\infty }\) weights on spaces of homogeneous type. J. Funct. Anal. 263, 3883–3899 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Iida, T., Sato, E., Sawano, Y., Tanaka, H.: Weighted norm inequalities for multilinear fractional operators on Morrey spaces. Stud. Math. 205, 139–170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282, 219–231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lacey, M., Moen, K., Pérez, C., Torres, R.: Sharp weighted bounds for fractional integral operators. J. Funct. Anal. 259, 1073–1097 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lerner, A., Pérez, C.: A new characterization of the Muckenhoupt \(A_p\) weights through an extension of the Lorentz-Shimogaki theorem. Indiana Univ. Math. J. 56, 2697–2722 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  16. Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nakamura, S.: Generalized weighted Morrey spaces and classical operators. Math. Nachr. 289, 2235–2262 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Olsen, P.: Fractional integration, Morrey spaces and Schrödinger equation. Comm. Partial Differ. Equ. 20, 2005–2055 (1995)

    Article  MATH  Google Scholar 

  19. Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350, 56–72 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tanaka, H.: Two-weight norm inequalities on Morrey spaces. Ann. Acad. Sci. Fenn. Math. 40, 773–791 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author is supported by Grant-in-Aid for JSPS Research Fellow (17J01766), Japan Society for the Promotion of Science. The second author is partially supported by Grand-in-Aid for Scientific Research (C) (16K05209), Japan Society for the Promotion of Science. The third author is supported by Grant-in-Aid for Scientific Research (C) (15K04918), Japan Society for the Promotion of Science.

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Authors and Affiliations

  1. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Shohei Nakamura & Yoshihiro Sawano

  2. Research and Support Center on Higher Education for the hearing and Visually Impaired, National University Corporation Tsukuba University of Technology, Kasuga 4-12-7, Tsukuba City, Ibaraki, 305-8521, Japan

    Hitoshi Tanaka

Authors
  1. Shohei Nakamura
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  2. Yoshihiro Sawano
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  3. Hitoshi Tanaka
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Correspondence to Shohei Nakamura.

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Nakamura, S., Sawano, Y. & Tanaka, H. The Fractional Operators on Weighted Morrey Spaces. J Geom Anal 28, 1502–1524 (2018). https://doi.org/10.1007/s12220-017-9876-2

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  • DOI: https://doi.org/10.1007/s12220-017-9876-2

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