Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem

Let $(M^n,g)$ be a complete, connected, noncompact Riemannian manifold, and let $f\ge0$ be a proper $C^2$ weight. We assume polynomial bounds on $f$, $|\nabla f|$, and $Δf$ (Assumptions H1-H3), with growth exponent $α>0$. We study the drift Laplacian $L_f=Δ-\langle\nabla f,\nabla\cdot\rangle$ on $L^