2009/03/27 by Der‐Chen Chang, Dachun Yang, Chang, Der-Chen +3
Mathematics · #Advanced Harmonic Analysis Research #Holomorphic and Operator Theory #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.0903.4725
Let p∈(0, 1]. In this paper, the authors prove that a sublinear operator T (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces Hp(\mathbb Rn×\mathbb Rm) to some quasi-Banach space \mathcal B if and only if T maps all (p, 2, s1, s2)-atoms into uniformly bounded elements of \mathcal B. Here s1≥\lfloor n(1/p-1)\rfloor and s2≥\lfloor m(1/p-1)\rfloor. As usual, \lfloor n(1/p-1)\rfloor denotes the maximal integer no more than n(1/p-1). Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space Lp(\mathbb Rn×\mathbb Rm) with some p>1 or the Hardy space Hp(\mathbb Rn×\mathbb Rm) with some p≤1 but near 1 to the Lebesgue space Lq(\mathbb Rn×\mathbb Rm) with some q>1.