Kôzô Yabuta

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 we extend a result given by Chen et al. (J. Math. Anal. Appl. 276:691-708, 2002). They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spaces F˙pqα(Rn) to Lp(Rn). 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=ϕ(|y|)y/|y|}⊂Rn×(Rn∖{0}). To prove our result, we discuss a characterization of the homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences.MSC:42B20, 42B25, 47G10.

On the boundedness of fractional type Marcinkiewicz integral operators

We show that a broad family of fractional type Marcinkiewicz integral operators with the kernel belonging to L1(Sn−1) is bounded from the Triebel-Lizorkin space Fα pq(R) to Lebesgue space Lp(Rn) , which improves some known results significantly. This is done by exploiting a local but more general fractional version of Littlewood-Paley g -function. Mathematics subject classification (2010): Primary 42B20; Secondary 42B25, 47G10.