On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals - Chen, Yanping; Tian, Tian (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 517.9Let $Ω$ be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere. For $m\in \mathbb N,$ let $b\in L_{\rm loc}^1(\mathbb R^n)$ and let the higher-order commutator of the Marcinkiewicz integral $μ_{Ω,b}^m$ be defined by \begin{gather*}μ_{Ω,b}^{m}(f)(x)=\left(\,\int\limits_{0}^{∞}\,\left|\,\,\int\limits_{|x-y| ≤ t} \frac{Ω(x-y)}{|x-y|^{n-1}}[b(x)-b(y)]^{m}f(y) dy\right|^{2} \frac{d t}{t^{3}}\right)^{\frac{1}{2}}.\end{gather*} We establish a sparse domination of $μ_{Ω,b}^{m}$ for $Ω\in {\rm Lip}(\mathbb{S}^{n-1})$. Moreover, we also give Bloom-type characterizations of the two-weighted boundedness of the higher-order commutators $μ_{Ω,b}^{m}, μ_{Ω,α,b}^{*,m}$, and $μ_{Ω,S,b}^{m}$, where the higher-order commutators $ μ_{Ω,α,b}^{*,m}$ and $μ_{Ω,S,b}^{m}$ are defined, respectively, by \begin{gather*}μ_{Ω,α,b}^{*,m}(f)(x)=\left(\,\,\iint\limits_{\mathbb{R}_{+}^{n+1}}\left(\frac{t}{t+|x-y|}\right)^{nα}\left|\,\int\limits_{|y-z| ≤ t} \frac{Ω(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}},\quad α>1,\end{gather*} and \begin{gather*}μ_{Ω,S,b}^{m}(f)(x)=\left(\,\,\,\iint\limits_{|x-y|