Obtaining upper bounds of heat kernels from lower bounds

We show that a near-diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on-diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off-diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near-diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. (C) 2007 Wiley Periodicals, Inc.