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# A note on $H^p_w$-boundedness of Riesz transforms and $\theta$-Calderón-Zygmund operators through molecular characterization

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 \cite{LLY} (J. Math. Anal. Appl. 301 (2005), 394--400) established that the Riesz transforms $R_j, j=1, 2,...,n$, are bounded on $H^p_w(\mathbb R^n)$. 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 \cite{LLY}, consists in passing from atoms to all functions in $H^p_w(\mathbb R^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.
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https://hal.archives-ouvertes.fr/hal-00602760
Contributor : Luong Dang Ky <>
Submitted on : Saturday, January 14, 2012 - 12:50:09 AM
Last modification on : Thursday, May 3, 2018 - 3:32:06 PM
Document(s) archivé(s) le : Sunday, April 15, 2012 - 2:21:02 AM

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• HAL Id : hal-00602760, version 3
• ARXIV : 1106.4724

### Citation

Luong Dang Ky. A note on $H^p_w$-boundedness of Riesz transforms and $\theta$-Calderón-Zygmund operators through molecular characterization. Analysis in Theory and Applications, Springer Verlag (Germany), 2011, 14 p. ⟨hal-00602760v3⟩

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