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# Duality theory of weighted Hardy spaces with arbitrary number of parameters

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## Abstract

In this paper, we use the discrete Littlewood-Paley-Stein analysis to get the duality result of the weighted product Hardy space for arbitrary number of parameters under a rather weak condition on the product weight w∈A ∞ (ℝ n 1 ×⋯×ℝ n k ). We will show that for any k≥2,(H w p (ℝ n 1 ×⋯×ℝ n k )) * =CMO w p (ℝ n 1 ×⋯×ℝ n k ) (a generalized Carleson measure), and obtain the boundedness of singular integral operators on BMO w . Our theorems even when the weight function w=1 extend the H 1 -BMO duality of Chang-R.Fefferman for the non-weighted two-parameter Hardy space H 1 (ℝ n ×ℝ m ) to H p (ℝ n 1 ×⋯×ℝ n k ) for all 0<p≤1 and our weighted theory extends the duality result of Krug-Torchinsky on weighted Hardy spaces H w p (ℝ n ×ℝ m ) for w∈A r (ℝ n ×ℝ m ) with 1≤r≤2 and r/2<p≤1 to H w p (ℝ n 1 ×⋯×ℝ n k ) with w∈A ∞ (ℝ n 1 ×⋯×ℝ n k ) for all 0<p≤1.
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... The generalized Carleson measure space CMO p was first introduced in [29]. It has been proved in [30] that the duality of multiparameter Hardy space H p ðℝ n 1 × ℝ n 2 Þ is CMO p ðℝ n 1 × ℝ n 2 Þ. We refer the readers to [26,31] for the recent development of generalized Carleson measure spaces. ...
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