Boundedness of sublinear operators on product Hardy spaces and its application

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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 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 .