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Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces

Published online by Cambridge University Press:  20 November 2018

Yongsheng Han ,
Ming-Yi Lee  and
Chin-Cheng Lin
Yongsheng Han
Affiliation:
Department of Mathematics, Auburn University, Auburn, AL 36849-5310, U.S.A.e-mail: hanyong@mail.auburn.edu
Ming-Yi Lee
Affiliation:
Department of Mathematics, National Central University, Chung-Li, Taiwan 320, Republic of Chinae-mail: mylee@math.ncu.edu.tw; clin@math.ncu.edu.tw
Chin-Cheng Lin
Affiliation:
Department of Mathematics, National Central University, Chung-Li, Taiwan 320, Republic of Chinae-mail: mylee@math.ncu.edu.tw; clin@math.ncu.edu.tw
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Abstract

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In this article, we establish a new atomic decomposition for $f\,\in \,L_{w}^{2}\,\bigcap \,H_{w}^{p}$, where the decomposition converges in $L_{w}^{2}$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L_{w}^{2}$ and $0\,<\,p\,\le \,1$, we obtain (i) if $T$ is uniformly bounded in $L_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$; (ii) if $T$ is uniformly bounded in $H_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded on $H_{w}^{p}$; (iii) if $T$ is bounded on $H_{w}^{p}$, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$.

Keywords

42B2542B30Ap weightsatomic decompositionCalderón reproducing formulaweighted Hardy spaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 303 - 314
Copyright
Copyright © Canadian Mathematical Society 2012

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