If $T$ is an differential operator of order $k$ from $\mathbb{C}$-vector bundle $E$ to $F$ over a compact differential manifold $X$.
Question: how can we extend it to a continuous linear map between the Sobolev spaces $W^s(X,E)$ and $W^{s-k}(X,E)$ for all integer $s$ ? Here, the Sobolev spaces $W^s(X,E)$ are the Hilbert spaces, also the completion of smooth sections of E w.r.t the Sobolev $s$-norm. When $s$ is negative, it is the space of corresponding distributions or just the dual of $W^{-s}(X,E)$.
I try to use the norm of $W^s(X,E)$ given by fourier transformation and partition of unity, but I went to a very complicated integration formula which is hard to estimate.
oops, luckily, we can extend the bounded operator $T$ on smooth sections to a bounded operator on sobolev spaces, given $D(T)$ is dense in $W^s(X,E)$. So suffice to show $T$ is bounded on normed space of smooth sections. Now it is easy! use partition of unity, since the cover is finite, we reduce to the $\mathbb{R}^n$ case.