## Obtaining Upper . . . From Lower Bounds (2007)

by
Alexander Grigoryan
,
Jiaxin Hu
,
Ka-sing Lau

### BibTeX

@MISC{Grigoryan07obtainingupper,

author = {Alexander Grigoryan and Jiaxin Hu and Ka-sing Lau},

title = {Obtaining Upper . . . From Lower Bounds},

year = {2007}

}

### OpenURL

### Abstract

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.