Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces

Abstract

Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-\gamma /t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set. Let $A$ be a nonnegative self-adjoint operator on sections of a finite-rank Euclidean vector bundle over a doubling metric measure space, satisfying ultracontractivity, Davies-Gaffney estimates (equivalently, finite speed of propagation for the wave equation) and a pointwise local Weyl law. If a weighted integrated observability inequality holds on a measurable set $\omega$, for a fixed horizon $T\in (0,+\infty ]$ and an admissible weight $h$, then, for every $0<\kappa <\frac{1}{2}$, $$h(t)\le A_{T,\kappa }\exp \left(-\kappa \frac{\mathcal{L}(\omega {)}^2}{t}\right),0<t<T,$$ where $\mathcal{L}(\omega )$ is the essential maximal distance to $\omega$, replaced by any finite radius when $\mathcal{L}(\omega )=+\infty$. Thus, for $h(t)=e^{-\gamma /t}$, necessarily $\gamma \ge \mathcal{L}(\omega {)}^2/2$. This settles, with the optimal threshold, the maximal-distance lower bound for the infinite-time constant left open in earlier work. In the control-norm convention, the fast-control rate is at least $\mathcal{L}(\omega {)}^2/4$, recovering Miller's bound. The proof rests on the spectral packet $(\cosh (r\sqrt{A})-1)e^{-tA}$. A pointwise Weyl law gives its sharp lower growth, while finite propagation speed and a weak-kernel Kannai transmutation formula make it exponentially small on $\omega$. Without kernel continuity or compact resolvent, we develop pointwise spectral measures and weak wave kernels. The framework covers Laplace-type operators on compact Riemannian manifolds, coupled heat systems, Schrödinger operators on ${\mathbb{R}}^d$, equiregular sub-Laplacians and Grushin models, and ${\delta }^{\prime }$-coupled Laplacians on metric graphs.