Multilinear BMO Estimates for the Commutators of Multilinear Fractional Maximal and Integral Operators on the Product Generalized Morrey Spaces

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Ferit Gurbuz, Multilinear BMO Estimates for the Commutators of Multilinear Fractional Maximal and Integral Operators on the Product Generalized Morrey Spaces, Int. J. Anal. Appl., 17 (4) (2019), 596-619.

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Ferit Gurbuz

Abstract

In this paper, we establish multilinear BMO estimates for commutators of multilinear fractional maximal and integral operators both on product generalized Morrey spaces and product generalized vanishing Morrey spaces, respectively. Similar results are still valid for commutators of multilinear maximal and singular integral operators.

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  1. X. N. Cao and D. X. Chen, The boundedness of Toeplitz-type operators on vanishing Morrey spaces, Anal. Theory Appl., 27 (2011), 309-319.
  2. D. X. Chen, J. Chen and S. Mao, Weighted Lp estimates for maximal commutators of multilinear singular integrals, Chin. Ann. Math., 34B(6) (2013), 885-902.
  3. X. Chen and Q. Y. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362 (2010), 355-373.
  4. R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.
  5. Z. W. Fu, Y. Lin and S. Z. Lu, λ-Central BMO estimates for commutators of singular integral operators with rough kernel, Acta Math. Sin. (Engl. Ser.), 24 (2008), 373-386.
  6. L. Grafakos, On multilinear fractional integrals, Studia. Math., 102 (1992), 49-56.
  7. L. Grafakos and R. H. Torres, Multilinear Calder ´on-Zygmund theory, Adv. Math., 165 (2002), 124-164.
  8. L. Grafakos and R. H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J., 51 (2002), 1261-1276.
  9. L. Grafakos and R. H. Torres, On multilinear singular integrals of Calder ´on-Zygmund type, in: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial), in: Publ. Mat., vol. Extra, 2002, 57-91.
  10. M. Giaquinta, Multiple integrals in the calculus of variations and non-linear elliptic systems. Princeton, New Jersey: Princeton Univ. Press, 1983.
  11. F. Gurbuz, Weighted Morrey and Weighted fractional Sobolev-Morrey Spaces estimates for a large class of pseudodifferential operators with smooth symbols, J. Pseudo-Differ. Oper. Appl., 7(4) (2016), 595-607.
  12. F. Gurbuz, Sublinear operators with rough kernel generated by Calderon-Zygmund operators and their commutators on generalized Morrey spaces, Math. Notes, 101(3-4) (2017), 429-442.
  13. F. Gurbuz, Some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, Canad. Math. Bull., 60(1) (2017), 131-145.
  14. F. Gurbuz, Multi-sublinear operators generated by multilinear fractional integral operators and local Campanato space estimates for commutators on the product generalized local Morrey spaces, Adv. Math. (China), 47(6) (2018), 855-880.
  15. F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14 (1961), 415-426.
  16. T. Karaman, Boundedness of some classes of sublinear operators on generalized weighted Morrey spaces and some applications
  17. [Ph.D. thesis], Ankara University, Ankara, Turkey, 2012 (in Turkish).
  18. C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1-15.
  19. Y. Lin, Strongly singular Calderon-Zygmund operator and commutator on Morrey type spaces, Acta Math. Sin. (Engl. Ser.), 23(11) (2007), 2097-2110.
  20. T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo (1991), 183-189.
  21. K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math., 60 (2009), 213-238.
  22. C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.
  23. B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274.
  24. D. K. Palagachev and L. G. Softova, Singular integral operators, Morrey spaces and fine regularity of solutions to PDE's, Potential Anal., 20 (2004), 237-263.
  25. Z. Y. Si and S. Z. Lu, Weighted estimates for iterated commutators of multilinear fractional operators, Acta Math. Sin. (Engl. Ser.), 28(9) (2012), 1769-1778.
  26. L. G. Softova, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.), 22(3) (2006), 757-766.
  27. M. E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Volume 81 of Math. Surveys and Monogr. AMS, Providence, R.I., 2000.
  28. C. Vitanza, Functions with vanishing Morrey norm and elliptic partial differential equations, in: Proceedings of Methods of Real Analysis and Partial Differential Equations, Capri, pp. 147-150. Springer (1990).
  29. J. Xu, Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures, Sci. China (Series A), 50 (2007), 361-376.
  30. X. Yu and X. X. Tao, Boundedness of multilinear operators on generalized Morrey spaces, Appl. Math. J. Chinese Univ., 29(2) (2014), 127-138.
  31. R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, vol. 43 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1977.