As far as I know, the following fact must be published somewhere, and I would like to find a reference.
The Mellin transform is defined by $f\mapsto\frac1{\sqrt{2\pi}}\int_0^\infty f(x)x^{s-1}\,dx$. The classical fact is that it is a unitary operator from $L^2(0, \infty)$ to the unweighted $L^2$-space on the critical line $\{\Re s=\frac12\}$. Also it is a unitary operator from the Hardy space $H^2$ on the half-plane $\{\Re z>0\}$ to the weighted $L^2$-space on the critical line with weight $e^{\pi t}+e^{-\pi t}$ if we write $s=\frac12+it$ for $s$ on the critical line; this is the fact I am interested in. To apply the Mellin transform to a function $f\in H^2$, one takes the above integral on $(0, \infty)$; the formula can be applied to functions that decrease exponentially near zero and infinity, such functions form a dense set.