We prove a certain inequality for a subsolution of the heat equation
associated with a regular Dirichlet form. As a consequence of this inequality, we
obtain various interesting comparison inequalities for heat semigroups and heat
kernels, which can be used for obtaining pointwise estimates of heat kernels. As
an example of application, we present a new method of deducing sub-Gaussian
upper bounds of the heat kernel from on-diagonal bounds and tail estimates