The uniform quantitive weighted boundedness of fractional Marcinkiewicz integral and its commutator
Huoxiong Wu, Lin Wu
TL;DR
The paper addresses the problem of obtaining uniform quantitative weighted bounds for the fractional Marcinkiewicz integral $\mu_{Ω,β}$ and its commutator as the fractional parameter $β$ approaches zero. It develops a dyadic Fourier-multiplier based decomposition, proves a sparse domination of a localized model $\widetilde{μ}_{Ω,β}$ by a fractional sparse operator, and then transfers these bounds to $μ_{Ω,β}$ and $μ_{Ω,β}^b$ via summation and a Cauchy integral argument. The main contributions are uniform weighted estimates for $μ_{Ω,β}$ on $A_{p,q}$ weights with explicit exponents depending on the regime of $β$ (0<β<1/2 and 1/2≤β<n), and analogous uniform bounds for the commutator $μ_{Ω,β}^b$ with $b\in BMO$, which recover the known $β→0^+$ results for the classical Marcinkiewicz operator and its commutator. The results extend fractional Marcinkiewicz theory to the weighted setting and provide a robust sparse-domination framework enabling limit transitions and refined weighted control in harmonic analysis.
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 $β\rightarrow 0^+$.