The boundedness of Marcinkiewicz integral with variable kernel

In this article, we study the fractional Marcinkiewicz integral with variable kernel defined by where 0 < α ≤ 2. We first prove that μΩ,α is bounded from L^2n/n+α(Rⁿ) to L²(Rⁿ) without any smoothness assumption on the kernel Ω. Then we show that, if the kernel Ω satisfies a class of Dini condition, μΩ,α is bounded from H^p(Rⁿ) (p ≤ 1) to H^q(Rⁿ), where 1/q = 1/p - α/2n. As corollary of the above results, we obtain the L^p - L^q (1 < p < 2) boundedness of this fractional Marcinkiewicz integral.