关键词: Generalized higher commutator of Marcinkiewicz integral, Hardy space, Lipschitz space

Abstract:

Let $\vec{b}=(b_{1},\cdots ,b_{m})$ be a finite family of locally
integrable functions. Then, we introduce generalized higher commutator of Marcinkiwicz integral as follows:
$$\mu_{\Omega}^{\vec{b}}(f)(x)=\Bigl(\int_{0}^{\infty}
|F_{\Omega,t}^{\vec{b}}(f)(x)|^{2}\frac{{\rm d}t}{t}\Bigr)^{1/2},$$ where
$$F_{\Omega,t}^{\vec{b}}(f)(x)=\frac{1}{t}\int_{|x-y|\leq t}
\frac{\Omega(x-y)}{|x-y|^{n-1}}\prod\limits_{j=1}^{m}(b_{j}(x)-b_{j}(y))f(y){\rm d}y.$$
When $b_{j}\in\dot{\Lambda}_{\beta_{j}}, 1\leq j\leq m,$
$0<\beta_{j}<1, \sum\limits_{j=1}^{m}\beta_{j}=\betais homogeneous of degree zero and satisfies the cancelation condition, we prove that $\mu_{\Omega}^{\vec{b}}$ is bounded from
$L^{p}({\Bbb R}^{n})$ to $L^{s}({\Bbb R}^{n}),$ where
$1Moreover, if $\Omega$ also satisfies some L^q-Dini condition, then
$\mu_{\Omega}^{\vec{b}}$ is bounded from $L^{p}({\Bbb R}^{n})$ to
$\dot{F}^{\beta,\infty}_{p}({\Bbb R}^{n})$ and on certain Hardy spaces.
The article extends some known results.

Key words: Generalized higher commutator of Marcinkiewicz integral, Hardy space, Lipschitz space