Maybe my question is just a matter of knowing the right equivalent definition.

The question is whether there is some relation between

$ H^p(D^2)$, defined as the space made of the analytic functions on the $2$-disk for which

$$\sup_{r\in]0,1[}\int|f(re^{i\theta})|^pd\theta<\infty$$

and the space $H^p(\mathbb R^2)$ which is defined for example as containing the functions such that $\sup_{t>0}|P_t*f|(x)\in L^p$, where is the Poisson kernel or some similar (but you can take $P_t(x)=t^{-2}P(x/t)$ and $P(x)$ some Schwartz function with nonzero mean).

The confusion of notation is just in this post, since (apparently) the books/people treating one kind of spaces never care about the other kind: are there any exceptions to this rule?

Thanks for the help!

Explorations in Harmonic Analysisor Hoepfner'sHardy spaces, its variants and applications. – Willie Wong Jan 18 '12 at 21:24