Davies, E. B. Heat kernel bounds for second order elliptic operators on Riemannian manifolds. (English) Zbl 0648.58037 Am. J. Math. 109, 545-569 (1987). In this paper, the author investigates the relationship between ultracontractive bounds on heat kernels, weighted Sobolev inequalities, and logarithmic Sobolev inequalities. He gives conditions which imply these bounds for a wide class of second order elliptic operators on manifolds and applies this result to Laplace-Beltrami operators on manifolds with cusps, and to operators on a manifold whose coefficients become degenerate or infinite on the boundary of the manifold. This extends an earlier paper of the author and B. Simon, J. Funct. Anal. 59, 335-395 (1984; Zbl 0568.47034). Reviewer: B.Helffer Cited in 16 Documents MSC: 58J35 Heat and other parabolic equation methods for PDEs on manifolds 53C20 Global Riemannian geometry, including pinching Keywords:ultracontractive bounds on heat kernels; weighted Sobolev inequalities; logarithmic Sobolev inequalities; second order elliptic operators; Laplace-Beltrami operators Citations:Zbl 0568.47034 PDFBibTeX XMLCite \textit{E. B. Davies}, Am. J. Math. 109, 545--569 (1987; Zbl 0648.58037) Full Text: DOI