The Fourier transform is defined for tempered distributions. For these distributions, the test functions are those functions decreasing more quickly at $\pm \infty$ than $|x|^{-n}$ for all n.
In contrast, the Mellin transform is defined for functions $(0,\infty) \to \Bbb R$. So what are the corresponding "tempered distributions" here? Is the idea that the functions are decreasing faster than $|\log x|^{-n}$ for all n? Or is there something more subtle to it?
Also, does this space of test functions (and its continuous dual space of distributions) have a name?