I have asked nearly the same question on Math Overflow, but it is maybe too low level.
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's book, Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and all its elements have zero mean.
I am looking for a (possibly sharp) Banach space $X$ of the following form: $$ \|u\|_{X}=\sum_{|\alpha|\leq k} \|(1+|x|)^N\partial^\alpha u\|_{L^p}, $$ $p\in[1,\infty]$, such that a function $u\in X$ belongs to $\mathcal H^1(\mathbb R^n)$ if and only if $\int_{\mathbb R^n} f=0$.
If polynomial weights are not sufficient, then it would be fine to have exponential weights, etc... A related question: is any Schwartz function with zero mean also in $\mathcal H^1(\mathbb R^n)$?
I wonder whether one can take $X=W^{k,1}(\mathbb R^n)$ for $k$ large enough, that is, $p=1$ and $N=0$.
What I know is that any function in $L^q(\mathbb R^n)$, $q>1$ with compact support and zero mean belongs to $\mathcal H^1(\mathbb R^n)$ (and the $L^q$ assumption can be relaxed to $L\log L$). I had a quick look at Stein's book and Grafakos' book, but I did not find anything like what I am looking for.
I have the suspect that assuming only decay of $u$ of any kind ($k=0$) is not sufficient, and one needs also the decay of some of its derivatives, but I could be wrong. That would be an answer for the question on MO.
Edit. It is clear that the main problem is at low frequencies. A weighted $L^p$ bound on $u$ would ensure that the Fourier transform is bounded and regular close to $\xi=0$. I am looking for spaces that have no other conditions on the low frequencies of $u$, this is why I am requiring $X$ to be a weighted Sobolev spaces.