关键词: Marcinkiewicz integral, rough kernel, uniform estimates, $A_p$ weight

Abstract: This paper considers the following Marcinkiewicz type integrals $$\mu_{\Omega,\beta }f(x) = \left ( \int_{0}^{\infty } \left | \int_{| x-y |\le t }^{} \frac{\Omega (x-y)}{| x-y|^{n-1-\beta }} f(y){\rm d}y \right | ^{2}\frac{{\rm d}t}{t^3} \right )^{{1}/{2} },\quad 0<\beta<n,$$ which can be regarded as an extension of the classical Marcinkiewicz integral $\mu_\Omega$ introduced by Stein in [Trans Amer Math Soc, 88(1958): 159-172], where $\Omega$ is a homogeneous function of degree zero on $\mathbb{R}^n$ with mean value zero in the unit sphere $S^{n-1}$. Under the assumption that $\Omega\in L^{\infty}(S^{n-1})$, the authors establish the $L^q$-estimate and weak $(1,1)$ type estimate as well as the corresponding weighted estimates for $\mu_{\Omega,\beta}$ with $1<q<\infty$ and $0<\beta<{(q-1)n}/{q}$. Moreover, the bounds do not depend on $\beta$ and the strong $(q,q)$ type and weak $(1,1)$ type estimates for the classical Marcinkiewicz integral $\mu_{\Omega}$ can be recovered from the above estimates of $\mu_{\Omega,\beta}$ when $\beta\to 0$.

Key words: Marcinkiewicz integral, rough kernel, uniform estimates, $A_p$ weight