The continuity of multilinear integral operators on Morrey spaces



Abstract

In this paper, the continuity of some multilinear integral operators on Morrey spaces are obtained. The operators contain singular integral operators, Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operators.


Keywords

multilinear operator; singular integral operator; Littlewood-Paley operator; Marcinkiewicz operator; Bochner-Riesz operator; Morrey space; BMO

1. Cohen J., A sharp estimate for a multilinear singular integral on ℝ^n, Indiana Univ. Math. J. 30 (1981), 693-702.
2. Cohen J., Gosselin J., On multilinear singular integral operators on ℝ^n, Studia Math. 72 (1982), 199-223.
3. Cohen J., Gosselin J., A BMO estimate for multilinear singular integral operators, Illinois J. Math. 30 (1986), 445-465.
4. Coifman R., Meyer Y., Wavelets, Calderón-Zygmund and multilinear operators, Cambridge Studies in Advanced Math. 48, Cambridge University Press, Cambridge 1997.
5. Coifman R., Rochberg R., Weiss G., Factorization theorems for Hardy spaces in several variables, Ann of Math. 103 (1976), 611-635.
6. Ding Y., Lu S.Z., Weighted boundedness for a class rough multilinear operators, Acta Math. Sinica 17 (2001), 517-526.
7. Fan D., Lu S.Z., Yang D.C., Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica 14 (1998), 625-634.
8. Di Fazio G., Ragusa M.A., Commutators and Morrey spaces, Boll. Un. Mat. I tal. (7)5—A (1991), 323-332.
9. Di Fazio G., Ragusa M.A., Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Func. Anal. 112 (1993), 241-256.
10. Liu L.Z., Weighted weak type (H^1, L ^1) estimates for commutators of Littlewood-Paley operator, Indian J. Math. 45 (2003), 71-78.
11. Liu L.Z., Weighted weak type estimates for commutators of Littlewood - Paley operator, Ja­pan. J. Math. 29 (2003), 1-13.
12. Liu L.Z., Continuity for multilinear Marcinkiewicz operators on certain Hardy spaces, Geor­gian Math. J. 10 (2003), 717-728.
13. Liu L.Z., The continuity of commutators on Triebel-Lizorkin spaces, Integral Equations and Operator Theory 49(1) (2004), 65-76.
14. Liu L.Z., Multilinear integral operators and mean oscillation, Proc. Indian Acad. Sci. (Math. Sci.) 114 (3) (2004), 235-251.
15. Liu L.Z., Sharp endpoint inequality for multilinear Littlewood-Paley operator, Kodai Math. J. 27 (2004), 134-143.
16. Liu L.Z., Lu S.Z., Weighted weak type inequalities for maximal commutators of Bochner-Riesz operator, Hokkaido Math. J. 32 (2003), 85-99.
17. Lu S.Z., Four lectures on real H^p spaces, World Scientific, River Edge, NI,1995.
18. Mizuhara T., Boundedness of some classical operators on generalized Morrey spaces, in "Harmonic Analysis", Proceedings of a conference held in Sendai, Japan, 1990, 183-189.
19. Peetre J., On the theory of L^{p,λ}-spaces, J. Funct. Anal. 4 (1969), 71-87.
20. Stein E.M., Harmonic Analysis: real variable methods orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 1993.
21. Torchinsky A., The real variable methods in harmonic analysis, Pure and Applied Math. 123, Academic Press, New York, 1986.
22. Torchinsky A., Wang S., A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235-243.
23. Wu B.S., Liu L.Z., A sharp estimate for multilinear Bochner-Riesz operator, Studia Sci. Math. Hungarica 42 (1) (2005), 47-59.
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Published : 2022-07-18


LiuL. (2022). The continuity of multilinear integral operators on Morrey spaces. Annales Mathematicae Silesianae, 20, 19-30. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14066

Lanzhe Liu  lanzheliu@263.net
College of Mathematics, Changsha University of Science and Technology, China  China



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