度量测度空间上热半群加权可观测性的最优几何障碍
Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces
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中文总结 AI 辅助
研究度量测度空间上热半群加权可观测性，通过谱包\((\cosh(r\sqrt A)-1)e^{-tA}\)等方法，证明热方程加权积分可观测性不等式中\(e^{-\gamma/t}\)尺度由观测集几何决定，确定最优阈值及快速控制率。
中文摘要 AI 辅助
热方程的加权积分可观测性不等式通常涉及形式为\(e^{-\gamma/t}\)的小时间因子。我们证明此尺度并非卡尔曼或谱方法的人为产物，而是由观测集的几何结构所决定。设\(A\)为加倍度量测度空间上有限秩欧几里得向量丛截面的非负自伴算子，满足超压缩性、戴维斯 - 加夫尼估计等条件。若加权积分可观测性不等式在可测集\(\omega\)上成立，对于固定的\(T\in(0,+\infty]\)和允许权重\(h\)，则有\(h(t)\leq A_{T,\kappa}\exp\left(-\kappa\frac{\mathcal{L}(\omega)^2}{t}\right)\)，\(0<t<T\)。这确定了早期工作中未解决的无限时间常数的最大距离下界的最优阈值。在控制范数约定下，快速控制率至少为\(\mathcal{L}(\omega)^2/4\)，恢复了米勒的界。证明基于谱包\((\cosh(r\sqrt A)-1)e^{-tA}\)。逐点魏尔定律给出其尖锐的低增长，有限传播速度和弱核卡奈变换公式使其在\(\omega\)上指数级小。我们在没有核连续性或紧预解式的情况下，发展了逐点谱测度和弱波核。该框架涵盖了紧致黎曼流形上的拉普拉斯型算子、耦合热系统、\(\mathbb{R}^d\)上的薛定谔算子、等正则次拉普拉斯算子和格鲁辛模型，以及度量图上的\(\delta'\) - 耦合拉普拉斯算子。
英文摘要
Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-γ/t}$. 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 $ω$, for a fixed horizon $T\in(0,+\infty]$ and an admissible weight $h$, then, for every $0<κ<\frac{1}{2}$, $$ h(t)\leq A_{T,κ}\exp\left(-κ\frac{\mathcal{L}(ω)^2}{t}\right),\qquad 0<t<T, $$ where $\mathcal{L}(ω)$ is the essential maximal distance to $ω$, replaced by any finite radius when $\mathcal{L}(ω)=+\infty$. Thus, for $h(t)=e^{-γ/t}$, necessarily $γ\geq\mathcal{L}(ω)^2/2$. 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 $\mathcal{L}(ω)^2/4$, 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 $ω$. 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 $δ'$-coupled Laplacians on metric graphs.
