Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel
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 &...
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| Date: | 2014 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2014
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2156 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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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