On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals

Abstract

UDC 517.9

Let $\Omega$ 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 $\mu_{\Omega,b}^m$ be defined by  $$\begin{gather*}\mu_{\Omega,b}^{m}(f)(x)=\left(\,\int\limits_{0}^{\infty}\,\left|\,\,\int\limits_{|x-y| \leq t} \frac{\Omega(x-y)}{|x-y|^{n-1}}[b(x)-b(y)]^{m}f(y) dy\right|^{2} \frac{d t}{t^{3}}\right)^{\frac{1}{2}}.\end{gather*}$$  We establish a sparse domination of $\mu_{\Omega,b}^{m}$ for $\Omega\in {\rm Lip}(\mathbb{S}^{n-1})$. Moreover, we also give Bloom-type characterizations of the two-weighted boundedness of the higher-order commutators $\mu_{\Omega,b}^{m}, \mu_{\Omega,\alpha,b}^{*,m}$, and $\mu_{\Omega,S,b}^{m}$, where  the higher-order commutators $\mu_{\Omega,\alpha,b}^{*,m}$ and $\mu_{\Omega,S,b}^{m}$ are defined, respectively, by  $$\begin{gather*}\mu_{\Omega,\alpha,b}^{*,m}(f)(x)=\left(\,\,\iint\limits_{\mathbb{R}_{+}^{n+1}}\left(\frac{t}{t+|x-y|}\right)^{n\alpha}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}},\quad \alpha>1,\end{gather*}$$  and  $$\begin{gather*}\mu_{\Omega,S,b}^{m}(f)(x)=\left(\,\,\,\iint\limits_{|x-y|<t}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}}.\end{gather*}$$