## Taiwanese Journal of Mathematics

### BOUNDEDNESS OF OPERATORS ON HARDY SPACES

#### Abstract

In [1], the author provided an example which shows that there is a linear functional bounded uniformly on all atoms in $H^1(\mathbb{R}^n)$, and it can not be extended to a bounded functional on $H^1(\mathbb{R}^n)$. In this note, we first give a new atomic decomposition, where the decomposition converges in $L^2(\mathbb{R}^n)$ rather than only in the distribution sense. Then using this decomposition, we prove that for $0 \lt p \leq 1$, $T$ is a linear operator which is bounded on $L^{2}(\mathbb{R}^n)$, then $T$ can be extended to a bounded operator from $H^{p}(\mathbb{R}^n)$ to $L^{p}(\mathbb{R})$ if and only if $T$ is bounded uniformly on all $(p,2)$-atoms in $L^{p}(\mathbb{R}^n)$. A similar result from $H^{p}(\mathbb{R}^n)$ to $H^{p}(\mathbb{R}^n)$ is also obtained. These results still hold for the product Hardy space and Hardy space on spaces of homogeneous type.

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 2 (2010), 319-327.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405791

Digital Object Identifier
doi:10.11650/twjm/1500405791

Mathematical Reviews number (MathSciNet)
MR2655771

Zentralblatt MATH identifier
1209.42013

Subjects
Primary: 42B30: $H^p$-spaces

#### Citation

Zhao, Kai; Han, Yongsheng. BOUNDEDNESS OF OPERATORS ON HARDY SPACES. Taiwanese J. Math. 14 (2010), no. 2, 319--327. doi:10.11650/twjm/1500405791. https://projecteuclid.org/euclid.twjm/1500405791

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