Boundedness of sublinear operators on product Hardy spaces and its application

摘要

Let p ∈ (0, 1]. In this paper, the authors prove that a snblinoar 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(R~n × R~m to some quasi-Banach space B if and only if T maps all (p, 2, s_1, s_2)-atoms into uniformly bounded elements of B. Here s_1 ≥ [n( 1/p -1)] and s_2 ≥ [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 Calderon-Zygmund operators and Lipschitz functions from the Lebesgue space L~p(R~n × R~m) with some p > 1 or the Hardy space H~P(R~n × R~m) with some p ≤ 1 but near 1 to the Lebesgue space L~q(R~n × R~m) with some q>1.