Boundedness of Calderón-Zygmund operators on weighted product hardy spaces

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Abstract

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 Hpw(R×R)-Lpw (R2) 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 Hpw (R×R)-boundedness of T provided T*1(1) = T*2(1) = 0; that is, the reduced T1 theorem on Hpw(R×R). In order to show these two results, we demonstrate a new atomic decomposition of Hpw(R×R) ∩ L2w(R2), for which the series converges in L2w. 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 languageEnglish
Pages (from-to)115-133
Number of pages19
JournalJournal of Operator Theory
Volume72
Issue number1
DOIs
StatePublished - 2014

Keywords

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

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