Let $M$ be a smooth manifold and $D$ a first-order differential operator acting on sections of a smooth vector bundle $E$ over $M$.
Denote by $H^i(E)$ the $i^\text{th}$ Sobolev space of sections of $E$. For each $t\in [0,\infty)$, let $\psi(t)$ be an element of $H^2(E)$, so that $D(\psi(t))\in H^1(E)$. Suppose that $\psi(x,t)$ and $D\psi(x,t)$ are uniformly (in $x\in M$) bounded above by non-negative continuous function $C(t)$ and $C'(t)$ respectively, and that
$$\int_0^\infty C(t)dt\qquad\text{and}\qquad\int_0^\infty C'(t)dt$$
converge. Suppose also that the integral
$$\int_0^\infty\psi(x,t) dt$$
is an element of $H^1(E)$, so that it can be acted on by $D$.
Question: Is it true that
$$\int_0^\infty D\psi(x,t) dt = D\int_0^\infty \psi(x,t) dt?$$
I suspect this question might boil down to a simpler question about interchanging differentiation and integration for Sobolev spaces of functions on $\mathbb{R}^n$, although I've stated it more generally.
Thanks!