Isomorphism between Sobolev space and L^p

Question

Let $L_1$ be an elliptic PDE operator $L_1:W^{2,p}\rightarrow L^p$ and $L_2=e^fL_1$ where f is a bounded function. I proved $L_1$ is an isomorphism, can I conclude $L_2$ is an isomorphism?

Answer 1

Yes. For any bounded function $f$ the multiplication operator $T(g)=e^fg$ is an isomorphism of $L^p$ onto $L^p$, with its inverse being $T^{-1}(g)=e^{-f}g$. Since $L_2$ is the composition of two isomorphisms, it is an isomorphism.