Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

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 ≤ j ≤ m. We prove the following weighted inequalities for ω ∈ A _∞ 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 ω ∈ A ₁.