Polynomial estimates for solutions of parametric elliptic equations on complete manifolds

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