Journal of Zhejiang University SCIENCE A
ISSN 1673-565X(Print), 1862-1775(Online), Monthly

2007   Vol. 8   No. 12   p. 2037~2040

A note on the Marcinkiewicz integral operators on F_p^α,q*

ZHANG Chun-jie^†1, QIAN Rui-rui²

(¹Department of Mathematics, Zhejiang University, Hangzhou 310027, China)
(²Sci-Tech Section, Zhejiang Sci-Tech University, Hangzhou 210018, China)
^†E-mail: purezhang@zju.edu.cn
Received Mar. 2, 2007 revision accepted Apr. 11, 2007

Abstract: In this paper, we shall prove that the Marcinkiewicz integral operator μ_Ω, 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.

Key words: Marcinkiewicz integral, Triebel-Lizorkin spaces, Fourier transforms
doi:10.1631/jzus.2007.A2037             CLC number: O174.4

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