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Is there a connection between the Hardy Spaces on the unit disk and on $\mathbb R^n$?.

If so, can we use results from the Hardy Spaces on the unit disk to prove $(H^1)^* = \text{BMO}$?

Further, what is the most fruitful way to define $H^1$ on $\mathbb R^n$ but avoiding (bounded) distributions? I was writing something for a project and this would be the only point where I would use them. Of course, I could take the completion of the $C_c^\infty$ functions in the $H^1$-norm, but is this workable enough?

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    @Theo: Thanks! The duality proof looks a bit different from the one I know (from Grafakos' book). I will definitely read it completely.2011-03-06

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Adding my comment as an answer on the OP's request:

Have you looked at the original paper by Fefferman and Stein? [Read the masters!] I have read (parts of) it a long time ago and I think it uses singular integral methods (and Hardy-Littlewood maximal functions and the like) for that. Here's a link to it.