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

Abstract

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