In this article, we study the fractional Marcinkiewicz integral with variable kernel defined by
\[\mu_{\Omega,\alpha}(f)(x)=\bigg(\int_{0}^{\infty}\bigg|{\int_{|x-y|\leq t}}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}f(y)\,dy\bigg|^{2}\frac{dt}{t^{3-\alpha}}\bigg)^{1/2},\]
¶
where 0<α≤2. We first prove that μ
Ω,α is bounded from L
2n/n+α(ℝ
n) to L
2(ℝ
n) without any smoothness assumption on the kernel Ω. Then we show that, if the kernel Ω satisfies a class of Dini condition, μ
Ω,α is bounded from H
p(ℝ
n) (p≤1) to H
q(ℝ
n), where 1/q=1/p−α/2n. As corollary of the above results, we obtain the L
p−L
q (1