Fractional type Marcinkiewicz integral operators associated to surfaces

TL;DR

This paper studies the boundedness of fractional Marcinkiewicz integral operators linked to specific surfaces, extending previous results by weakening assumptions and characterizing function spaces involved.

Contribution

It extends prior boundedness results of fractional Marcinkiewicz operators to new surface cases with weaker conditions and characterizes related Triebel-Lizorkin spaces.

Findings

Boundedness of operators on Triebel-Lizorkin spaces established

Weakened assumptions compared to previous work

Characterization of homogeneous Triebel-Lizorkin spaces via lacunary sequences

Abstract

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.