For what it worth, in the terms of [divergent integrals][1], your transform can be rewritten as

$$\operatorname{reg} \int_1^\infty f(t)e^{-t s \omega _-}dt$$
Looks like some kind of analog of Fourier transform, if you ask me...


  [1]: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651