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?