Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces

Han, Yongsheng ; Lee, Ming-Yi ; Lin, Chin-Cheng

Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 303-314

Résumé

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 $L_{w}^{2}$ .

Détail

DOI : 10.4153/CMB-2011-072-7

Mots-clés : 42B25, 42B30, Ap weights, atomic decomposition, Calderón reproducing formula, weighted Hardy spaces