I am trying to understand the Mellin transform of $\phi(x) \mathbf{1}_{\leq T}(x)$ where $\phi$ is smooth and $\mathbf{1}_{\leq T}(x)$ is a function that is $1$ if $x\leq T$ and $0$ otherwise.
I would like to know two things: 1. Is it possible to get an expression in terms of $T$ and the Mellin transform of $\phi$? 2. If not, can we at least obtain a bound for it?
I have made an honest attempt trying to find the answer but I am quite lost at this and I would greatly appreciate any explanations/comments. Thank you very much.