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Received: 2007-03-02

Revision Accepted: 2007-04-11

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Journal of Zhejiang University SCIENCE A 2007 Vol.8 No.12 P.2037~2040

http://doi.org/10.1631/jzus.2007.A2037


A note on the Marcinkiewicz integral operators on Fpα,q


Author(s):  ZHANG Chun-jie, QIAN Rui-rui

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China; more

Corresponding email(s):   purezhang@zju.edu.cn

Key Words:  Marcinkiewicz integral, Triebel-Lizorkin spaces, Fourier transforms


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ZHANG Chun-jie, QIAN Rui-rui. A note on the Marcinkiewicz integral operators on Fpα,q[J]. Journal of Zhejiang University Science A, 2007, 8(12): 2037~2040.

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Abstract: 
In this paper, we shall prove that the marcinkiewicz integral operator μΩ, when its kernel Ω satisfies the L1-Dini condition, is bounded on the triebel-Lizorkin spaces. It is well known that the triebel-Lizorkin spaces are generalizations of many familiar spaces such as the Lebesgue spaces and the Sobolev spaces. Therefore, our result extends many known theorems on the marcinkiewicz integral operator. Our method is to regard the marcinkiewicz integral operator as a vector valued singular integral. We also use another characterization of the Triebel-Lizorkin space which makes our approach more clear.

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Reference

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[6] Chen, D.N., Chen, J.C., Fan, D.S., 2005. Rough singular integral operators on Hardy-Sobolev spaces. Appl. Math. J. Chin. Univ. Ser. B, 20(1):1-9.

[7] Ding, Y., Fan, D.S., Pan, Y.B., 2000. Lp-boundedness of Marcinkiewicz integrals with Hardy space function kernels. Acta. Math. Sinica, 16:593-600.

[8] Korry, S., 2004. A class of bounded operators on Sobolev spaces. Arch. Math., 82(1):40-50.

[9] Stein, E.M., 1958. On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Amer. Math. Soc., 88:430-466.

[10] Triebel, H., 1983. Theory of Function Spaces. Birkhäuser-Verlag, Basel-Boston, p.108.

[11] Xu, H., Chen, J.C., Ying, Y.M., 2003. A note on Marcinkiewicz integrals with H1 kernels. Acta. Math. Scientia, 23:133-138.

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