Abstract
This paper is a continuation of our previous work (Zhang and Chen, 2010b). Following the same general steps of the proof there, we make essential improvement on our previous theorem by recalculating a key inequality. Our result shows that the Marcinkiewicz integral, with a bounded radial function in its kernel, is still bounded on the Triebel-Lizorkin space.
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Al-Qassem, H.M., Cheng, L.C., Pan, Y., 2012. Boundedness of rough integral operators on Triebel-Lizorkin spaces. Publ. Mat., 56(2):261–277. [doi:10.5565/PUBLMAT_ 56212_01]
Chen, D.N., Chen, J.C., Fan, D.S., 2005. Rough singular integral operators on Hardy-Sobolev spaces. Appl. Math. J. Chin. Univ., 20(1):1–9. [doi:10.1007/s11766- 005-0030-8]
Chen, J.C., Zhang, C.J., 2008. Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces. J. Math. Anal. Appl., 337(2):1048–1052. [doi:10.1016/ j.jmaa.2007.04.026]
Chen, J.C., Fan, D.S., Ying, Y.M., 2002. Singular integral operators on function spaces. J. Math. Anal. Appl., 276(2):691–708. [doi:10.1016/S0022-247X(02)00419-5]
Chen, J.C., Fan, D.S., Ying, Y.M., 2003. Certain operators with rough singular kernels. Can. J. Math., 55(3): 504–532.
Chen, J.C., Jia, H.Y., Jiang, L.Y., 2005. Boundedness of rough oscillatory singular integral on Triebel-Lizorkin spaces. J. Math. Anal. Appl., 306(2):385–397. [doi:10. 1016/j.jmaa.2005.01.015]
Chen, Q.L., Zhang, Z.F., 2004. Boundedness of a class of super singular integral operators and the associated commutators. Sci. China Ser. A, 47(6):842–853. [doi:10.1360/03ys0099]
Chen, Y.P., Ding, Y., 2008. Rough singular integrals on Triebel-Lizorkin space and Besov space. J. Math. Anal. Appl., 347(2):493–501. [doi:10.1016/j.jmaa.2008.06.039]
Chen, Y.P., Zhu, K., 2014. Lp bounds for the commutators of oscillatory singular integrals with rough kernels. Abs. Appl. Anal., 2014: 393147.1-393147.8. [doi:10.1155/2014/393147]
Zhang, C.J., Chen, J.C., 2009. Boundedness of g-functions on Triebel-Lizorkin spaces. Taiwan. J. Math., 13(3): 973–981.
Zhang, C.J., Chen, J.C., 2010a. Boundedness of singular integrals and maximal singular integrals on Triebel-Lizorkin spaces. Int. J. Math., 21(2):157–168. [doi:10.1142/S0129167X10005982]
Zhang, C.J., Chen, J.C., 2010b. Boundedness of Marcinkiewicz integral on Triebel-Lizorkin spaces. Appl. Math. J. Chin. Univ., 25(1):48–54. [doi:10.1007/s11766-010-2086-3]
Zhang, C.J., Zhang, Y.D., 2013. Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces. Appl. Math. J. Chin. Univ., 28(1):90–100. [doi:10.1007/s11766-013-3033-x]
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Project supported by the National Natural Science Foundation of China (Nos. 11201103 and 11471288)
ORCID: Chun-jie ZHANG, http://orcid.org/0000-0002-5712-8764
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Zhang, Cj., Ren, Ff., Zhang, Yh. et al. Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces. Frontiers Inf Technol Electronic Eng 16, 654–657 (2015). https://doi.org/10.1631/FITEE.1500082
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DOI: https://doi.org/10.1631/FITEE.1500082