Fractional type Marcinkiewicz integral operators associated to surfaces

In this paper, we discuss the boundedness of the fractional type Marcinkiewicz integral operators associated to surfaces, and extend a result given by Chen, Fan and Ying in 2002. They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spaces $\dot F_{pq}^{\alpha}({\mathbb R}^n)$ to $L^p({\mathbb R}^n)$. Recently the second author, together with Xue and Yan, greatly weakened their assumptions. In this paper, we extend their results to the case where the operators are associated to the surfaces of the form $\{x=\phi(|y|)y/|y|\} \subset {\mathbb R}^n \times ({\mathbb R}^n \setminus \{0\})$. To prove our result, we discuss a characterization of the homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences.Comment: 27page

Triebel--Lizorkin space estimates for multilinear operators of sublinear operators

In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel--Lizorkin space. The operators include Littlewood--Paley operator and Marcinkiewicz operator.Comment: 15 pages, no figures, no table

Boundedness of rough integral operators on Triebel-Lizorkin spaces

We prove the boundedness of several classes of rough integral operators on Triebel-Lizorkin spaces. Our results represent improvements as well as natural extensions of many previously known results

A Note on a Class of Generalized Parabolic Marcinkiewicz Integrals along Surfaces of Revolution

In this article, certain sharp Lp estimates for a specific class of generalized Marcinkiewicz operators with mixed homogeneity associated to surfaces of revolution are established. By virtue of Yano’s extrapolation argument, beside these estimates, the Lp boundedness of the aforementioned operators under weaker assumptions on the kernels is confirmed. The obtained results in this article are fundamental extensions and improvements of numerous previously known results on parabolic generalized Marcinkiewicz integrals