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On the Boundedness of Integral Operators in Weighted Grand Morrey Spaces

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Abstract

We obtain boundedness criteria in terms of Muckenhoupt weights for the Hardy–Littlewood maximal operator and Riesz transforms in weighted grand Morrey spaces \(M^{p),q,\varphi}_w\). We also consider some structural properties of the spaces \(M^{p),q,\varphi}_w\). The spaces are defined, generally speaking, on spaces of homogeneous type. The results are new even in the case of a special function \(\varphi\).

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References

  1. D. R. Adams, Morrey Spaces (Birkhäuser, Cham, 2015).

    Book  Google Scholar 

  2. R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières (Springer, Berlin, 1971), Lect. Notes Math. 242.

    Book  Google Scholar 

  3. X. Duoandikoetxea and M. Rosental, “Extension and boundedness of operators on Morrey spaces from extrapolation techniques and embeddings,” J. Geom. Anal. 28 (4), 3081–3108 (2018).

    Article  MathSciNet  Google Scholar 

  4. X. Duoandikoetxea and M. Rosental, “Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights,” J. Funct. Anal. 279 (8), 108687 (2020); arXiv: 1910.13902 [math.FA].

    Article  MathSciNet  Google Scholar 

  5. A. Fiorenza, B. Gupta, and P. Jain, “The maximal theorem for weighted grand Lebesgue spaces,” Stud. Math. 188 (2), 123–133 (2008).

    Article  MathSciNet  Google Scholar 

  6. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups (Princeton Univ. Press, Princeton, NJ, 1982), Math. Notes 28.

    MATH  Google Scholar 

  7. N. Fujii, “Weighted bounded mean oscillation and singular integrals,” Math. Japon. 22 (5), 529–534 (1978).

    MathSciNet  MATH  Google Scholar 

  8. J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam, 1985), North-Holland Math. Stud. 116.

    Book  Google Scholar 

  9. K.-P. Ho, “Definability of singular integral operators on Morrey–Banach spaces,” J. Math. Soc. Japan 72 (1), 155–170 (2020).

    Article  MathSciNet  Google Scholar 

  10. S. V. Hruščev, “A description of weights satisfying the \(A_\infty \) condition of Muckenhoupt,” Proc. Am. Math. Soc. 90 (2), 253–257 (1984).

    MathSciNet  Google Scholar 

  11. T. Hytönen, C. Pérez, and E. Rela, “Sharp reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous type,” J. Funct. Anal. 263 (12), 3883–3899 (2012).

    Article  MathSciNet  Google Scholar 

  12. V. M. Kokilashvili and A. N. Meskhi, “Weighted extrapolation in Iwaniec–Sbordone spaces. Applications to integral operators and approximation theory,” Proc. Steklov Inst. Math. 293, 161–185 (2016) [transl. from Tr. Mat. Inst. Steklova 293, 167–192 (2016)].

    Article  MathSciNet  Google Scholar 

  13. V. Kokilashvili and A. Meskhi, “The boundedness of sublinear operators in weighted Morrey spaces defined on spaces of homogeneous type,” in Function Spaces and Inequalities, Ed. by P. Jain and H.-J. Schmeisser (Springer, Singapore, 2017), Springer Proc. Math. Stat. 206, pp. 193–211.

    Article  MathSciNet  Google Scholar 

  14. V. Kokilashvili, A. Meskhi, and H. Rafeiro, “Boundedness of sublinear operators in weighted grand Morrey spaces,” Math. Notes 102 (5–6), 664–676 (2017) [transl. from Mat. Zametki 102 (5), 721–735 (2017)].

    Article  MathSciNet  Google Scholar 

  15. V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-standard Function Spaces, Vol. 2: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces (Birkhäuser/Springer, Basel, 2016).

    Book  Google Scholar 

  16. V. Kokilashvili, A. Meskhi, and M. A. Ragusa, “Weighted extrapolation in grand Morrey spaces and applications to partial differential equations,” Rend. Lincei, Mat. Appl. 30 (1), 67–92 (2019).

    Article  MathSciNet  Google Scholar 

  17. Y. Komori and S. Shirai, “Weighted Morrey spaces and a singular integral operator,” Math. Nachr. 282 (2), 219–231 (2009).

    Article  MathSciNet  Google Scholar 

  18. Y. Liu and W. Yuan, “Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces,” Czech. Math. J. 67 (3), 715–732 (2017).

    Article  MathSciNet  Google Scholar 

  19. R. A. Macías and C. Segovia, “Lipschitz functions on spaces of homogeneous type,” Adv. Math. 33 (3), 257–270 (1979).

    Article  MathSciNet  Google Scholar 

  20. A. Meskhi, “Maximal functions, potentials and singular integrals in grand Morrey spaces,” Complex Var. Elliptic Eqns. 56 (10–11), 1003–1019 (2011).

    Article  MathSciNet  Google Scholar 

  21. A. Meskhi and Y. Sawano, “Density, duality and preduality in grand variable exponent Lebesgue and Morrey spaces,” Mediterr. J. Math. 15 (3), 100 (2018).

    Article  MathSciNet  Google Scholar 

  22. R. Ch. Mustafayev, “On boundedness of sublinear operators in weighted Morrey spaces,” Azerb. J. Math. 2 (1), 66–79 (2012).

    MathSciNet  MATH  Google Scholar 

  23. S. Nakamura and Y. Sawano, “The singular integral operator and its commutator on weighted Morrey spaces,” Collect. Math. 68 (2), 145–174 (2017).

    Article  MathSciNet  Google Scholar 

  24. S. Nakamura, Y. Sawano, and H. Tanaka, “The fractional operators on weighted Morrey spaces,” J. Geom Anal. 28 (2), 1502–1524 (2018).

    Article  MathSciNet  Google Scholar 

  25. H. Rafeiro, “A note on boundedness of operators in grand grand Morrey spaces,” in Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary Volume, Ed. by A. Almeida, L. Castro, and F.-O. Speck (Birkhäuser, Basel, 2013), pp. 349–356.

    Article  MathSciNet  Google Scholar 

  26. M. Rosental and H.-J. Schmeisser, “The boundedness of operators in Muckenhoupt weighted Morrey spaces via extrapolation techniques and duality,” Rev. Mat. Complut. 29 (3), 623–657 (2016).

    Article  MathSciNet  Google Scholar 

  27. N. Samko, “Weighted Hardy and singular operators in Morrey spaces,” J. Math. Anal. Appl. 350 (1), 56–72 (2009).

    Article  MathSciNet  Google Scholar 

  28. N. Samko, “On a Muckenhoupt-type condition for Morrey spaces,” Mediterr. J. Math. 10 (2), 941–951 (2013).

    Article  MathSciNet  Google Scholar 

  29. Y. Sawano and H. Tanaka, “Predual spaces of Morrey spaces with non-doubling measures,” Tokyo J. Math. 32 (2), 471–486 (2009).

    Article  MathSciNet  Google Scholar 

  30. S. Shi, Z. Fu, and F. Zhao, “Estimates for operators on weighted Morrey spaces and their applications to nondivergence elliptic equations,” J. Inequal. Appl. 2013, 390 (2013).

    Article  MathSciNet  Google Scholar 

  31. H. Tanaka, “Two-weight norm inequalities on Morrey spaces,” Ann. Acad. Sci. Fenn., Math. 40 (2), 773–791 (2015).

    Article  MathSciNet  Google Scholar 

  32. D. Wang, J. Zhou, and W. Chen, “Another characterizations of Muckenhoupt \(A_p\) class,” Acta Math. Sci., Ser. B, Engl. Ed. 37 (6), 1761–1774 (2017).

    Article  MathSciNet  Google Scholar 

  33. C. T. Zorko, “Morrey space,” Proc. Am. Math. Soc. 98, 586–592 (1986).

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We are grateful to the referee for useful remarks.

Funding

This work was supported by the Shota Rustaveli National Science Foundation of Georgia, contract no. FR-18-2499.

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Authors and Affiliations

  1. Andrea Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Tamarashvili Str. 6, Tbilisi, 0186, Georgia

    V. M. Kokilashvili & A. N. Meskhi

  2. Georgian Technical University, Kostava Str. 77, Tbilisi, 0160, Georgia

    A. N. Meskhi

Authors
  1. V. M. Kokilashvili
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  2. A. N. Meskhi
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Correspondence to V. M. Kokilashvili.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 312, pp. 203–215 https://doi.org/10.4213/tm4134.

Translated by I. Nikitin

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Kokilashvili, V.M., Meskhi, A.N. On the Boundedness of Integral Operators in Weighted Grand Morrey Spaces. Proc. Steklov Inst. Math. 312, 194–206 (2021). https://doi.org/10.1134/S0081543821010119

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