Polynomial estimates for solutions of parametric elliptic equations on complete manifolds

Abstract

Let $P : \CI(M; E) \to \CI(M; F)$ be an order $\mu$ differential operator
with smooth enough coefficients $a = (a^{[0]}, a^{[1]}, \ldots, a^{[\mu]})$.
Let $P_k := P : H^{s_0 + k +\mu}(M; E) \to H^{s_0 + k}(M; F)$. We prove polynomial
norm estimates for the solution $P_0^{-1}f$ of
the form $$\|P_0^{-1}f\|_{H^{s_0 + k + \mu}(M; E)} \le C \sum_{q=0}^{k} \,
\| P_0^{-1} \|^{q+1} \,\|a \|_{W^{|s_0|+k}}^{q} \, \| f \|_{H^{s_0 + k - q}},$$
(thus in higher order Sobolev spaces, which amounts also to a parametric regularity result).
In particular, $P_k$ is invertible, if $a$ is smooth enough.
The assumptions are that $E, F \to M$ are Hermitian vector bundles and that
$M$ is a complete manifold satisfying the Fr\'echet Finiteness Condition (FFC),
which was introduced in (Kohr and Nistor, Annals of Global Analysis and Geometry, 2022).
These estimates are useful for uncertainty
quantification, since the coefficient $a$ can be regarded as a vector valued random
variable. We use these results to prove integrability of the norm $\|P_k^{-1}f\|$
of the solution of $P_k u = f$ with respect to suitable Gaussian measures.