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On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators

On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral... AbstractLet T be a bilinear Calderón–Zygmund singular integral operator and let T*{T^{*}} be its corresponding truncated maximal operator. For any b∈BMO⁡(ℝn){b\in\operatorname{BMO}({\mathbb{R}^{n}})} andb→=(b1,b2)∈BMO⁡(ℝn)×BMO⁡(ℝn){\vec{b}=(b_{1},b_{2})\in\operatorname{BMO}({\mathbb{R}^{n}})\times%\operatorname{BMO}({\mathbb{R}^{n}})}, let Tb,j*{T^{*}_{b,j}} (j=1,2{j=1,2}) and Tb→*{T^{*}_{\vec{b}}} be the commutators in the j-th entry and the iterated commutators of T*{T^{*}}, respectively.In this paper, for all 1<p1,p2<∞{1<p_{1},p_{2}<\infty}, 1p=1p1+1p2{\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}}, we show that Tb,j*{T^{*}_{b,j}} and Tb→*{T^{*}_{\vec{b}}} are compact operators from Lp1⁢(w1)×Lp2⁢(w2){L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})} to Lp⁢(vw→){L^{p}(v_{\vec{w}})} if b,b1,b2∈CMO⁡(ℝn){b,b_{1},b_{2}\in\operatorname{CMO}(\mathbb{R}^{n})} and w→=(w1,w2)∈Ap→{\vec{w}=(w_{1},w_{2})\in A_{\vec{p}}}, vw→=w1p/p1⁢w2p/p2{v_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}}. Here CMO⁡(ℝn){\operatorname{CMO}(\mathbb{R}^{n})} denotes the closure of 𝒞c∞⁢(ℝn){\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})} in the BMO⁡(ℝn){\operatorname{BMO}(\mathbb{R}^{n})} topology and Ap→{A_{\vec{p}}} is the multiple weights class. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators

Wang, Shifen; Xue, Qingying
Forum Mathematicum , Volume 34 (2): 16 – Mar 1, 2022
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Publisher
de Gruyter
Copyright
© 2022 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2020-0357
Publisher site
See Article on Publisher Site

Abstract

AbstractLet T be a bilinear Calderón–Zygmund singular integral operator and let T*{T^{*}} be its corresponding truncated maximal operator. For any b∈BMO⁡(ℝn){b\in\operatorname{BMO}({\mathbb{R}^{n}})} andb→=(b1,b2)∈BMO⁡(ℝn)×BMO⁡(ℝn){\vec{b}=(b_{1},b_{2})\in\operatorname{BMO}({\mathbb{R}^{n}})\times%\operatorname{BMO}({\mathbb{R}^{n}})}, let Tb,j*{T^{*}_{b,j}} (j=1,2{j=1,2}) and Tb→*{T^{*}_{\vec{b}}} be the commutators in the j-th entry and the iterated commutators of T*{T^{*}}, respectively.In this paper, for all 1<p1,p2<∞{1<p_{1},p_{2}<\infty}, 1p=1p1+1p2{\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}}, we show that Tb,j*{T^{*}_{b,j}} and Tb→*{T^{*}_{\vec{b}}} are compact operators from Lp1⁢(w1)×Lp2⁢(w2){L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})} to Lp⁢(vw→){L^{p}(v_{\vec{w}})} if b,b1,b2∈CMO⁡(ℝn){b,b_{1},b_{2}\in\operatorname{CMO}(\mathbb{R}^{n})} and w→=(w1,w2)∈Ap→{\vec{w}=(w_{1},w_{2})\in A_{\vec{p}}}, vw→=w1p/p1⁢w2p/p2{v_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}}. Here CMO⁡(ℝn){\operatorname{CMO}(\mathbb{R}^{n})} denotes the closure of 𝒞c∞⁢(ℝn){\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})} in the BMO⁡(ℝn){\operatorname{BMO}(\mathbb{R}^{n})} topology and Ap→{A_{\vec{p}}} is the multiple weights class.

Journal

Forum Mathematicum – de Gruyter

Published: Mar 1, 2022

Keywords: Bilinear maximal Calderón–Zygmund singular integral operators; commutator; compactness; 42B20; 42B25

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