中文摘要：

该文讨论了在加倍测度度量空间中,热核估计的上界与下界的关系.若相应的狄氏型满足局部性条件,则可以由近对角下界估计推出上对角估计.和此前这类问题的研究工作相比,该文给出了更加一般性的结果.

英文摘要：

This paper shows the relationship between lower bounds and upper bounds of the heat kernel on metric spaces with doubling measure. If in addition the Dirichlet form is local, then a near-diagonal lower bound implies an on-diagonal upper bound. This paper gives the upper estimate in balls and then extends it to full spaces. Compared with previous work, the conclusion of this paper not only contains former result but reveals a more general relationship between the lower and upper bounds.