Heat kernel estimates and the relative compactness of perturbations by potentials

Abstract

We consider a self-adjoint non-negative operator $H$ in a Hilbert space ${\mathsf{L}}^2(X,\mathrm{d}\mu )$. We assume that the semigroup $({\mathrm{e}}^{-tH}{)}_{t>0}$ is defined by an integral kernel, $p$, which allows an estimate of the form $p(t,x,x)\le F_1(x)F_2(t)$ for all $(x,t)\in X\times {\mathbb{R}}_{+}$; we refer to $F_1$ as the \emph{control function}. We show that such an estimate leads to rather satisfying abstract results on relative compactness of perturbations of $H$ by potentials. It came as a surprise to us, however, that such an estimate holds for the Laplace-Beltrami operator on \emph{any} Riemannian manifold. In particular, using a domination principle, one can deduce from the latter fact a very general result on the relative compactness of perturbations by potentials of the Bochner Laplacian associated with a Hermitian bundle $(E,h^E,{\nabla }^E)$ over an arbitrary Riemannian manifold $(M,g)$; in fact, only quantities of order zero in $g$ enter in the estimates. We extend this result to weighted Riemannian manifolds, where under lower curvature bounds on the $\alpha$-Bakry-Émery tensor one can construct quite explicit control functions, and to any weighted graph, where the control function is expressed in terms of the vertex weight function.