  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.
