2016/06/02 by Jochen Brüning, Brüning, Jochen, Batu Güneysu +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP) #math.DG #math.FA #math.SP
paper · pdf · doi:10.48550/arxiv.1606.00651
arxiv created 2016/06/02 · openalex publication_date 2016/06/02 · arxiv updated 2016/06/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We consider a self-adjoint non-negative operator H in a Hilbert space L2(X,\rm dμ). We assume that the semigroup (e-t H)t>0 is defined by an integral kernel, p, which allows an estimate of the form p(t,x,x)≤ F1(x)F2(t) for all (x,t)∈ X×\mathbbR+; we refer to F1 as the 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 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, hE,∇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 α-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.