Boundedness of Sublinear Operators on Product Hardy Spaces and Its Application

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