The boundedness of Marcinkiewicz integral with variable kernel

Abstract

In this article, we study the fractional Marcinkiewicz integral with variable kernel defined by $$\mu_{\Omega,\alpha}(f)(x)=\bigg(\int_{0}^{\infty}\bigg| {\int_{|x-y|\leq t}}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}f(y)\,dy\bigg|^{2} \frac{dt}{t^{3-\alpha}}\bigg)^{1/2}, %\\ %\quad0<\alpha\leq2. $$where $0<\alpha\leq2$. We first prove that $\mu_{\Omega,\alpha}$ is bounded from $L^{{2n}/{n+\alpha}}(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$ without any smoothness assumption on the kernel $\Omega$. Then we show that, if the kernel $\Omega$ satisfies a class of Dini condition, $\mu_{\Omega,\alpha}$ is bounded from $H^p(\mathbb{R}^n)$ ($p \le1$) to $H^q(\mathbb{R}^n)$, where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{2n}$. As corollary of the above results, we obtain the $L^p-L^q$ ($1< p<2$) boundedness of this fractional Marcinkiewicz integral.