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Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

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Ukrainian Mathematical Journal Aims and scope

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.

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Authors and Affiliations

  1. Mudanjiang Normal University, Mudanjiang, China

    Jianglong Wu

  2. University of Nova Gorica, Nova Gorica, Slovenia

    Qingguo Liu

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  1. Jianglong Wu
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  2. Qingguo Liu
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 4, pp. 538–550, April, 2014.

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Wu, J., Liu, Q. Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel. Ukr Math J 66, 602–616 (2014). https://doi.org/10.1007/s11253-014-0957-0

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  • DOI: https://doi.org/10.1007/s11253-014-0957-0

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