Boundedness of sublinear operators on product Hardy spaces and its application

Abstract

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 H^p(Rⁿ×R^m) 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 L^p(Rⁿ×R^m) with some p>1 or the Hardy space H^p(Rⁿ×R^m) with some p≤1 but near 1 to the Lebesgue space L^q(Rⁿ×R^m) with some q>1.