Given smooth vector bundles $E \to M$ and $F \to M$ over a smooth manifold $M$, a linear differential operator from $E$ to $F$ is an $\mathbb{R}$-linear sheaf homomorphism $D: \mathcal{E} \to \mathcal{F}$.
Given a smooth map $f: N \to M$, can we define a pulled back differential operator $f^*D: f^* \mathcal{E} \to f^*\mathcal{F}$? If not, under what conditions on the map $f$ or operator $D$ can we do so?
I believe that the answer is "no" for general smooth maps $f$ because $D$ is not a homomorphism of $\mathcal{O}_M$-modules, so I would not expect it to interact nicely with the pullback functor $f^*: \mathcal{O}_M\mathsf{Mod} \to \mathcal{O}_N\mathsf{Mod}$.
However, if $f$ is the inclusion of an open submanifold, then the pullback functor is just the inverse image of sheaves $f^*=f^{-1}$, so $f^*D$ makes sense and is just the restriction of $D$ to sections over $N$.
I would hope that if $f$ is the inclusion of a closed submanifold then $f^*D$ can be defined, but unfortunately in this situation $f^* \neq f^{-1}$ in general.