Let $M$ be a Riemannian manifold, and define the Sobolev spaces $H^k(M)$ to be the set of distributions $f$ on $M$ such that $Pf \in L^2(M)$ for all differential operators $P$ on $M$ of order less than or equal to $k$ with smooth coefficients.
If $F : M\rightarrow N$ is a diffeomorphism, must it hold that $H^k(M)$ and $H^k(N)$ are linearly isomorphic? If $F$ and all its derivatives are bounded, then composition with $F$ gives the desired isomorphism. Is there still a way to produce an isomorphism if this is not the case?