Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

  • Qingguo Liu
  • Jianglong Wu

Abstract

Let \( {\mu}_{\varOmega, \overrightarrow{b}} \) be a multilinear commutator generalized by the n-dimensional Marcinkiewicz integral with bounded kernel μ Ώ and let \( {b}_j\ \in Os{c_{\exp}}_{L^{r_j}} \) , 1 ≤ jm. We prove the following weighted inequalities for ωA and 0 < p < ∞: $$ {\begin{array}{cc}\hfill {\left\Vert {\mu}_{\varOmega }(f)\right\Vert}_{L^p\left(\omega \right)}\le C{\left\Vert M(f)\right\Vert}_{L^p\left(\omega \right)},\hfill & \hfill \left\Vert {\mu}_{\varOmega, \overrightarrow{b}}(f)\right\Vert \hfill \end{array}}_{L^p\left(\omega \right)}\le C{\left\Vert {M}_{L{\left( \log L\right)}^{1/r}}(f)\right\Vert}_{L^p\left(\omega \right)}. $$

The weighted weak L(log L)1/r -type estimate is also established for p =1 and ωA 1.

Published
25.04.2014
How to Cite
LiuQ., and WuJ. “Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals With Bounded Kernel”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, no. 4, Apr. 2014, pp. 538–550, http://umj.imath.kiev.ua/index.php/umj/article/view/2156.
Section
Research articles