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Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

Research paper by Guo Zhen Lu, Ya Yuan Xiao

Indexed on: 01 Feb '12Published on: 01 Feb '12Published in: Acta Mathematica Sinica, English Series



Abstract

The main purpose of this paper is to derive a new (p, q)-atomic decomposition on the multi-parameter Hardy space Hp(X1 × X2) for 0 < p0 < p ≤ 1 for some p0 and all 1 < q < ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq(X1 × X2) (for 1 < q < ∞) and Hardy space Hp(X1 × X2) (for 0 < p ≤ 1). As an application, we prove that an operator T, which is bounded on Lq(X1 × X2) for some 1 < q < ∞, is bounded from Hp(X1 × X2) to Lp(X1 × X2) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp(X1 × X2). The similar boundedness criterion from Hp(X1 × X2) to Hp(X1 × X2) is also obtained.