Title: Multilinear BMO Estimates for the Commutators of Multilinear Fractional Maximal and Integral Operators on the Product Generalized Morrey Spaces
Author(s): Ferit Gurbuz
Pages: 596-619
Cite as:
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.

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.

Full Text: PDF

 

References


  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  4. R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331. Google Scholar

  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. Google Scholar

  6. L. Grafakos, On multilinear fractional integrals, Studia. Math., 102 (1992), 49-56. Google Scholar

  7. L. Grafakos and R. H. Torres, Multilinear Calder´on-Zygmund theory, Adv. Math., 165 (2002), 124-164. Google Scholar

  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. Google Scholar

  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. Google Scholar

  10. M. Giaquinta, Multiple integrals in the calculus of variations and non-linear elliptic systems. Princeton, New Jersey: Princeton Univ. Press, 1983. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  15. F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14 (1961), 415-426. Google Scholar

  16. T. Karaman, Boundedness of some classes of sublinear operators on generalized weighted Morrey spaces and some applications Google Scholar

  17. [Ph.D. thesis], Ankara University, Ankara, Turkey, 2012 (in Turkish). Google Scholar

  18. C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1-15. Google Scholar

  19. Y. Lin, Strongly singular Calderon-Zygmund operator and commutator on Morrey type spaces, Acta Math. Sin. (Engl. Ser.), 23(11) (2007), 2097-2110. Google Scholar

  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. Google Scholar

  21. K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math., 60 (2009), 213-238. Google Scholar

  22. C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166. Google Scholar

  23. B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. Google Scholar

  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. Google Scholar

  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. Google Scholar

  26. L. G. Softova, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.), 22(3) (2006), 757-766. Google Scholar

  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. Google Scholar

  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). Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar