The Davies Method Revisited For Heat Kernel Upper Bounds

We are concerned with heat kernels on metric measure spaces,which have been proved to be useful in math,physics,chemistry and biology.It is well-known that the function t?e^atplays an important role in analysis.No wonder that as a far-reaching generalization of the exponential function,the heat semigroup{e^-tA}_t?0,where A denotes some positive definite operator,plays similarly an indispensable role in modern mathematics and physics,not the least because it solves the corresponding heat equation u'+Au=0.If A acts in a function space,then the action of semigroup exp?-tA?is frequently given by an integral operator,whose kernel is called the heat kernel of A.Naturally,any knowledge of the heat kernel,say upper and lower bounds of estimates,can be useful in understanding various problems related to A and its spectrum including solutions to the heat equation,properties of the underlying space,etc.In particular,if the operator A is Markovian,that is,generates a Markov process?for example,this is the case if A is a second order differential operator?,then one can make use of the information about the heat kernel to help solving problems concerning the process itself.In general,a regular Dirichlet form on metric measure spaces canonically leads to the notion of the heat semigroup and the heat kernel,where we can define the latter either as the transition density of a Hunt process or as the integral kernel of the heat semigroup.According to whether the Dirichlet form is local,we have two classes of heat kernel estimates:local estimate and nonlocal estimate.An interesting question is what minimal assumptions that are equivalent to heat kernel bounds.For local heat kernel upper estimate,mathematicians have considered Poincar??? inequality,Faber-Krahn inequality,Sobolev inequality,volume doubling condition,etc.As for two-sided estimates,they also need elliptic Harnack inequality or just use parabolic Hanack inequality.To characterize nonlocal upper bound,we also need estimates on jump kernel.Mathematicians have used analytical and probabilistic approaches to characterize heat kernel bounds.Our approach is purely analytical.One well-known method,named after Davies,was introduced by E.B.Davies in 1987 to estimate heat kernel upper bounds on Riemannian manifold.In the same year,Carlen,Kusuoka and Stroock generalized it to the nonlocal case.Recently,Murugan and Saloff-Coste further improved it for strongly local Dirichlet forms by using Andres and Barlow's cutoff inequality on annuli.We apply the Davies method to prove that for any regular Dirichlet form on a metric measure space,an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound,a cutoff inequality on any two concentric balls,and the jump kernel upper bound,for any walk dimension.If in addition the jump kernel vanishes,that is,if the Dirichlet form is strongly local,we obtain sub-Gaussian upper bound.This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.