关键词: fractional type Marcinkiewicz integrals, commutators, Muckenhoupt weights, ${\rm BMO}$ spaces, Hardy spaces

Abstract: This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integrals $\mu_{\Omega, \beta}$ and the commutators $\mu_{\Omega, \beta}^b$ generated by $\mu_{\Omega, \beta}$ with $b\in L_{\rm loc}(\mathbb{R}^n)$ on weighted Hardy spaces. Under the assumption of that the homogeneous kernel $\Omega$ satisfies certain regularities, the authors obtain the boundedness of $\mu_{\Omega, \beta}$ from the weighted Hardy spaces $H^p_{\omega^p}(\mathbb{R}^n)$ to the weighted Lebesgue spaces $L^q_{\omega^q}(\mathbb{R}^n)$ for $n/(n+\beta)\le p\le 1$ with $1/q=1/p-\beta/n$, as well as the same $(H^p_{\omega^p}, L^q_{\omega^q})$-boudedness of $\mu_{\Omega, \beta}^b$ when $b$ belongs to $\mathcal{BMO}_{\omega^p, p}(\mathbb{R}^n)$, which is a non-trivial subspace of ${\rm BMO}(\mathbb{R}^n)$.

Key words: fractional type Marcinkiewicz integrals, commutators, Muckenhoupt weights, ${\rm BMO}$ spaces, Hardy spaces