Journal of Operator Theory

Volume 72, Issue 1, Summer 2014 pp. 115-133.

Boundedness of Calderon-Zygmund operators on weighted product Hardy spaces

Authors: Ming-Yi Lee
Author institution: Department of Mathematics, National Central University, Chung-Li, Taiwan 320, Republic of China

Summary: Let $T$ be a singular integral operator in Journé's class with regularity exponent $\varepsilon$, $w\in A_q$, $1\leqslant q<1+\varepsilon$, and $q/{(1+\varepsilon)}\lt p\leqslant 1$. We obtain the $H^p_w(\Bbb R\times \Bbb R)$-$L^p_w(\Bbb R^2)$ boundedness of $T$ by using R. Fefferman's `trivial lemma' and Journ\'e's covering lemma. Also, using the vector-valued version of the `trivial lemma' and Littlewood-Paley theory, we prove the $H^p_w(\Bbb R \times \Bbb R)$-boundedness of $T$ provided $T^*_{1}(1)=T^*_{2}(1)=0$; that is, the reduced $T1$ theorem on $H^p_w(\Bbb R \times \Bbb R)$. In order to show these two results, we demonstrate a new atomic decomposition of $H^p_w(\Bbb R\times \Bbb R)\cap L^2_w(\Bbb R^2)$, for which the series converges in $L^2_w$. Moreover, a fundamental principle that the boundedness of operators on the weighted product Hardy space can be obtained simply by the actions of such operators on all atoms is given.

DOI: http://dx.doi.org/10.7900/jot.2012nov06.1993
Keywords: Calderon-Zygmund operator, Littlewood-Paley theory, weighted product Hardy space

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