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Volume 34, Issue 3
Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators

Cuilan Wu, Yunjie Wang & Lisheng Shu

Anal. Theory Appl., 34 (2018), pp. 209-224.

Published online: 2018-11

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  • Abstract

$[b,T]$ denotes the commutator of generalized Calderón-Zygmund operators $T$ with Lipschitz function $b$, where $b ∈ \rm{Lip}_β(R^n)$, $(0<β≤1)$ and $T$ is a $θ(t)$−type Calderón-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by 

$$[b,T] f(x)=b(x)Tf(x)−T(bf)(x)=∫k(x,y)(b(x)−b(y))f(y)dy.$$

In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(ω^p)$ to $L^q(ω^q)$, and from $H\dot{K}^{α,p}_{q_1}(ω_1,ω^{q_1}_2)$ to $\dot{K}^{α,p}_{q_2}(ω_1,ω^{q_2}_2)$. The results extend and generalize the well-known ones in [7].

  • AMS Subject Headings

42B20, 42B25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-34-209, author = {}, title = {Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators}, journal = {Analysis in Theory and Applications}, year = {2018}, volume = {34}, number = {3}, pages = {209--224}, abstract = {

$[b,T]$ denotes the commutator of generalized Calderón-Zygmund operators $T$ with Lipschitz function $b$, where $b ∈ \rm{Lip}_β(R^n)$, $(0<β≤1)$ and $T$ is a $θ(t)$−type Calderón-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by 

$$[b,T] f(x)=b(x)Tf(x)−T(bf)(x)=∫k(x,y)(b(x)−b(y))f(y)dy.$$

In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(ω^p)$ to $L^q(ω^q)$, and from $H\dot{K}^{α,p}_{q_1}(ω_1,ω^{q_1}_2)$ to $\dot{K}^{α,p}_{q_2}(ω_1,ω^{q_2}_2)$. The results extend and generalize the well-known ones in [7].

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2017-0050}, url = {http://global-sci.org/intro/article_detail/ata/12836.html} }
TY - JOUR T1 - Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators JO - Analysis in Theory and Applications VL - 3 SP - 209 EP - 224 PY - 2018 DA - 2018/11 SN - 34 DO - http://doi.org/10.4208/ata.OA-2017-0050 UR - https://global-sci.org/intro/article_detail/ata/12836.html KW - Commutator, Lipschitz function, weighted hardy space, Herz space. AB -

$[b,T]$ denotes the commutator of generalized Calderón-Zygmund operators $T$ with Lipschitz function $b$, where $b ∈ \rm{Lip}_β(R^n)$, $(0<β≤1)$ and $T$ is a $θ(t)$−type Calderón-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by 

$$[b,T] f(x)=b(x)Tf(x)−T(bf)(x)=∫k(x,y)(b(x)−b(y))f(y)dy.$$

In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(ω^p)$ to $L^q(ω^q)$, and from $H\dot{K}^{α,p}_{q_1}(ω_1,ω^{q_1}_2)$ to $\dot{K}^{α,p}_{q_2}(ω_1,ω^{q_2}_2)$. The results extend and generalize the well-known ones in [7].

Cuilan Wu, Yunjie Wang & Lisheng Shu. (1970). Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators. Analysis in Theory and Applications. 34 (3). 209-224. doi:10.4208/ata.OA-2017-0050
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