Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

Abstract

Let ${\mu }_{\Omega ,\vec{b}}$ be the multilinear commutator generalized by ${\mu }_{\Omega }$, the $n$-dimensional Marcinkiewicz integral with the bounded kernel, and $b_{j}\in \Osc_{\exp L^{r_{j}}}(1\le j\le m)$. In this paper, the following weighted inequalities are proved for $\omega \in A_{\infty }$ and $0<p<\infty$, $$\Vert {\mu }_{\Omega }(f){\Vert }_{L^p(\omega )}\le C\Vert M(f){\Vert }_{L^p(\omega )},\Vert {\mu }_{\Omega ,\vec{b}}(f){\Vert }_{L^p(\omega )}\le C\Vert M_{L(\log L{)}^{1/r}}(f){\Vert }_{L^p(\omega )}.$$ The weighted weak $L(\log L{)}^{1/r}$ -type estimate is also established when $p=1$ and $\omega \in A_1$.

In 2012, it is accepted by Ukrainian Mathematical Journal. And there are 13 pages

References in corpus (1)

• Weighted Endpoint Estimates for Multilinear Commutators of Marcinkiewicz Integrals