Boundedness of sublinear operators on product Hardy spaces and its application

概要

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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(Rn×Rm) to some quasi-Banach space ℬ if and only if T maps all (p,2,s1,s2)-atoms into uniformly bounded elements of ℬ. Here s1≥⌊n(1⁄p−1)⌋ and s2≥⌊m(1⁄p−1)⌋. As usual, ⌊n(1⁄p−1)⌋ 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(Rn×Rm) with some p>1 or the Hardy space Hp(Rn×Rm) with some p≤1 but near 1 to the Lebesgue space Lq(Rn×Rm) with some q>1.