( might not be a research-level question... if it violates any term of this website I will delete it ASAP )

I'm wondering if there is any way to show the boundedness/unboundedness of operators like $(\partial_x + f )^{-1}$ or $(H\partial_x +f)^{-1}$, such as in $L^2 (\mathbb{R})$. (Here $f$ is a bounded smooth function, but possibly we need more assumptions...? ) Could anyone give me a clue or tell me if there is any literature to follow?

Thanks in advance!