## Abstract

Let T be a singular integral operator in Journé's class with regularity exponent ε, w ∈ A_{q}, 1 ≤ q < 1 + ε, and q/(1 + ε) < p ≤ 1. We obtain the H^{p}_{w}(R×R)-L^{p}_{w} (R^{2}) 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 H^{p}_{w} (R×R)-boundedness of T provided T*_{1}(1) = T*_{2}(1) = 0; that is, the reduced T1 theorem on H^{p}_{w}(R×R). In order to show these two results, we demonstrate a new atomic decomposition of H^{p}_{w}(R×R) ∩ L^{2}_{w}(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.

Original language | English |
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Pages (from-to) | 115-133 |

Number of pages | 19 |

Journal | Journal of Operator Theory |

Volume | 72 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

## Keywords

- Calderón-Zygmund operator
- Littlewood-Paley theory
- Weighted product Hardy space