Let $E \to M$ be a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not necessarily compact). Let $H : \Gamma(E) \to \Gamma(E)$ be a differential operator with smooth coefficients such that its principal symbol is $g$, the Riemann tensor of $M$ (so $H$ is a generalized Laplacian). If $p : (0, \infty) \times M \to M$ is the natural projection, let $F = p^* E$ be the pull-back bundle. Let $u$ be a section in $F$ with distributional values such that $(\partial_t - H) u = 0$ in the distributional sense.
Are there ready-made tools available that would allow me to conclude that $u$ is smooth on $(0, \infty) \times M$?
The question reminds one of the concepts of hypoellipticity or parabolic regularity for functions. I could try to mimick these and produce analogous results for sections, but for sure I am not the first one to need them, so maybe they have already been obtained and I just don't know where to look for them (a Google search didn't help either). I also believe that this problem is not just some quick corollary of the similar results for functions.
(In my concrete problem $u$ is some $L^2 _{loc}$-integrable section. $(0, \infty)$ could very well be replaced with $\mathbb R$, of course, but in my work I need the heat semigroup later on.)
