The heat kernel on asymptotically hyperbolic manifolds

Upper and lower bounds on the heat kernel on complete Riemannian manifolds were obtained in a series of pioneering works due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. However, these estimates do not give a complete picture of the heat kernel for all times and all pairs of points - in particular, there is a considerable gap between available upper and lower bounds at large distances and/or large times. Inspired by the work of Davies-Mandouvalos onHn+1,we study heat kernel bounds on Cartan-Hadamard manifolds that are asymptotically hyperbolic in the sense of Mazzeo-Melrose. Under the assumption of no eigenvalues and no resonance at the bottom of the continuous spectrum, we show that the heat kernel on such manifolds iscomparableto the heat kernel on hyperbolic space of the same dimension (expressed as a function of timetand geodesic distancer),uniformlyfor allt is an element of(0,infinity)and allr is an element of[0,infinity).In particular our upper and lower bounds are uniformly comparable for all distances and all times. The corresponding statement for asymptoticallyEuclideanspaces is not known to hold, and as we argue in the last section, it is very unlikely to be true in that geometry. As an application, we show boundedness onL(p)of the Riesz transform backward difference (Delta-n2/4+lambda 2)-1/2,for lambda is an element of(0,n/2],on such manifolds, forpsatisfying|p-1-2-1|<lambda/n.For lambda=n/2(the standard Riesz transform backward difference Delta-1/2), this was previously shown by Lohoue in a more general setting.