Volume 27, Issue 3
A Note on $H^p_w$-Boundedness of Riesz Transforms and $\theta$-Calderόn-Zygmund Operators Through Molecular Characterization

Anal. Theory Appl., 27 (2011), pp. 251-264.

Published online: 2011-08

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• Abstract

Let $0 <p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang[11] established that the Riesz transforms $R_j$, $j = 1,2, \cdots ,n$, are bounded on $H^p_w(\mathbf{R}^n)$. In this note we extend this to the general case of weight $w$ in the Muckenhoupt class $A_\infty$ through molecular characterization. One difficulty, which has not been taken care in [11], consists in passing from atoms to all functions in $H^p_w(\mathbf{R}^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.

• Keywords

Muckenhoupt weight, Riesz transform, Calderón-Zygmund operator.

42B20, 42B25, 42B30

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@Article{ATA-27-251, author = {}, title = {A Note on $H^p_w$-Boundedness of Riesz Transforms and $\theta$-Calderόn-Zygmund Operators Through Molecular Characterization}, journal = {Analysis in Theory and Applications}, year = {2011}, volume = {27}, number = {3}, pages = {251--264}, abstract = {

Let $0 <p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang[11] established that the Riesz transforms $R_j$, $j = 1,2, \cdots ,n$, are bounded on $H^p_w(\mathbf{R}^n)$. In this note we extend this to the general case of weight $w$ in the Muckenhoupt class $A_\infty$ through molecular characterization. One difficulty, which has not been taken care in [11], consists in passing from atoms to all functions in $H^p_w(\mathbf{R}^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.

}, issn = {1573-8175}, doi = {https://doi.org/10.1007/s10496-011-0251-z}, url = {http://global-sci.org/intro/article_detail/ata/4598.html} }
TY - JOUR T1 - A Note on $H^p_w$-Boundedness of Riesz Transforms and $\theta$-Calderόn-Zygmund Operators Through Molecular Characterization JO - Analysis in Theory and Applications VL - 3 SP - 251 EP - 264 PY - 2011 DA - 2011/08 SN - 27 DO - http://doi.org/10.1007/s10496-011-0251-z UR - https://global-sci.org/intro/article_detail/ata/4598.html KW - Muckenhoupt weight, Riesz transform, Calderón-Zygmund operator. AB -

Let $0 <p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang[11] established that the Riesz transforms $R_j$, $j = 1,2, \cdots ,n$, are bounded on $H^p_w(\mathbf{R}^n)$. In this note we extend this to the general case of weight $w$ in the Muckenhoupt class $A_\infty$ through molecular characterization. One difficulty, which has not been taken care in [11], consists in passing from atoms to all functions in $H^p_w(\mathbf{R}^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.

Luong Dang Ky. (1970). A Note on $H^p_w$-Boundedness of Riesz Transforms and $\theta$-Calderόn-Zygmund Operators Through Molecular Characterization. Analysis in Theory and Applications. 27 (3). 251-264. doi:10.1007/s10496-011-0251-z
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