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

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

For what it worth, in the terms of divergent integrals, 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...