The uniform quantitive weighted boundedness of fractional Marcinkiewicz integral and its commutator

Abstract

Suppose that $\Omega \in L^{\infty}(\mathbb{S} ^{n-1})$ is homogeneous of degree zero with mean value zero. Then we consider a fractional type Marcinkiewicz integral operator $$\mu_{\Omega ,\beta }f(x) = \left ( \int_{0}^{\infty } \left | \int_{\left | x-y \right |\le t }^{} \frac{\Omega (x-y)}{\left | x-y \right |^{n-1-\beta } } f(y)dy \right | ^{2}\frac{dt}{t^3} \right )^{\frac{1}{2} },\quad 0<\beta<n.$$ Our main contribution is the quantitive weighted result of the classical Marcinkiewicz integral $\mu_{\Omega}$ proved by Hu and Qu [Math. Ineq. appl., 22(2019), 885-899] can be recovered from the quantitative weighted estimates of $\mu_{\Omega,\beta}$ in this paper when $\beta\to 0^+$. As inference, we also gives the uniform quantitive weighted bounds for the corresponding fractional commutators of $\mu_{\Omega,\beta}$ when $\beta \rightarrow 0^+$.