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 $${\mu }_{\Omega ,\beta }f(x)={\left({\int }_0^{\infty }{\left|{\int }_{|x-y|\le t}^{}\frac{\Omega (x-y)}{{\left|x-y\right|}^{n-1-\beta }}f(y)dy\right|}^2\frac{dt}{t^3}\right)}^{\frac{1}{2}},0<\beta <n.$$ 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 $\beta \to 0^{+}$.