BOUNDEDNESS OF CALDERÓN–ZYGMUND OPERATORS ON WEIGHTED PRODUCT HARDY SPACES

• Published in: Journal of operator theory 2014-08, Vol.72 (1), p.115-133, Article 115
• Main Author: LEE, MING-YI
• Format: Article
• Language: English
• Subjects: Dyadics; Mathematical functions; Mathematical integrals; Mathematical theorems; Rectangles; Urelements
• ISSN: 0379-4024; 1841-7744
• DOI: 10.7900/jot.2012nov06.1993

Let T be a singular integral operator in Journé's class with regularity exponent ε, w ∈ Aq, 1 ≤ q < 1 + ε, and q/(1 + ε) < p ≤ 1. We obtain the ${\mathrm{H}}_{\mathrm{w}}^{\mathrm{p}}(\mathrm{\mathbb{R}}\times \mathrm{\mathbb{R}})-{\mathrm{L}}_{\mathrm{w}}^{\mathrm{p}}\left({\mathrm{\mathbb{R}}}^{2}\right)$ boundedness of T by using R. Fefferman's "trivial lemma" and Journé's covering lemma. Also, using the vector-valued version of the "trivial lemma" and Littlewood–Paley theory, we prove the ${\mathrm{H}}_{\mathrm{w}}^{\mathrm{p}}(\mathrm{\mathbb{R}}\times \mathrm{\mathbb{R}})$-boundedness of T provided ${\mathrm{T}}_{1}^{*}\left(1\right)={\mathrm{T}}_{2}^{*}\left(1\right)=00$; that is, the reduced T1 theorem on ${\mathrm{H}}_{\mathrm{w}}^{\mathrm{p}}(\mathrm{\mathbb{R}}\times \mathrm{\mathbb{R}})$. In order to show these two results, we demonstrate a new atomic decomposition of ${\mathrm{H}}_{\mathrm{w}}^{\mathrm{p}}(\mathrm{\mathbb{R}}\times \mathrm{\mathbb{R}})\cap {\mathrm{L}}_{\mathrm{w}}^{2}\left({\mathrm{\mathbb{R}}}^{2}\right)$, for which the series converges in ${\mathrm{L}}_{\mathrm{w}}^{2}$. 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.