Weighted Estimates for Fractional Type Marcinkiewicz Integral Operators Associated to Surfaces

In this paper, we shall discuss the weighted boundedness of the fractional type Marcinkiewicz integral operators associated to surfaces, and extend a result given by Chen, Fan and Ying in 2002 and the one given by authors. They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spaces Ḟ pq (R) to L(R). Recently one of the author, together with Xue and Yan, greatly weakened their assumptions, and we further extended their results to the case where the operators are associated to the surfaces of the form {x = φ(|y|)y/|y|} ⊂ R × (R \ {0}). In this paper, we investigate weighted estimates for these operators. To prove our result, we discuss a characterization of the weighted homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences. 2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B25, 47G10