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

In this paper, we shall prove that the Marcinkiewicz integral operator #n, when its kernel Ω satisfies the L^1-Dini condition, is bounded on the Triehel-Lizorkin spaces. It is well known that the Triehel-Lizorkin spaces are generalizations of many familiar spaces such as the Lehesgue spaces and the Soholev 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 Triehel-Lizorkin space which makes our approach more clear.

In this paper, we shall prove that the Marcinkiewicz integral operator μ, when its kernel 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.