I'm trying to organize my knowledge about Hardy spaces, especially $\mathcal{H}^1$. In Stein's book "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" there is quite comprehensive description of the properties of the real Hardy spaces in $\mathbb{R}^n$. However, I have problems with understanding how these spaces behave on the $n$-dimensional torus $\mathbb{T}^n$.
So here go more precise questions:
In $\mathbb{R}^n$ we have the characterization $$ \|f\|_{\mathcal{H}^1(\mathbb{R}^n)} = \|f\|_{L^1(\mathbb{R}^n)} + \sum_{j=1}^n \|R_jf\|_{L^1(\mathbb{R}^n)}, $$ where $R_j$ is the Riesz transform given as $\mathcal{F}(R_jf)(\xi)=-i\frac{\xi_j}{|\xi|}\hat{f}(\xi)$. Would the same characterization work in $\mathbb{T}^n$? Am I correct that the Riesz transform would be then defined as $$ R_jf(x) = -i\sum_{m\in \mathbb{Z}^n}\frac{m_j}{|m|}e^{2\pi i m\cdot x}\hat{f}(m)? $$
Hardy spaces on the unit circle. There is another characterization of the Hardy spaces, defined on the unit circle $\mathbb{T}$ as the limit of the Hardy space on the unit disk: $\mathcal{H}^p(D)$ is the space of holomorphic functions defined on the open unit disk, such that $$ \sup_{0\leq r<1}\left(\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^p\mathrm{d}\theta \right)^{1/p} < \infty. $$ Then the space $\mathcal{H}^p(\mathbb{T})$ consists of all functions $\tilde{f}$ defined as $\tilde{f}(e^{it})=\lim_{r\to 1}f(re^{it})$ for $f\in \mathcal{H}^p(D)$. In this case we have the characterization \begin{equation}\label{n} f\in \mathcal{H}^1(\mathbb{T}) \; \text{ iff } \; f\in L^1(\mathbb{T}) \text{ and } \hat{f}(n)=0 \text{ for all } n<0.\tag{$\star$} \end{equation} What would be the equivalent of (\ref{n}) in the multi-dimensional case? Also, how this condition corresponds to the $\mathbb{R}^n$ case? The function from real Hardy space $\mathcal{H}^1(\mathbb{R}^n)$ must fulfill $$ \int_{\mathbb{R}^n} f(x)\mathrm{d}x = \hat{f}(0) = 0, $$ but condition (\ref{n}) does not relate to $n=0$.