  This paper aims at proving the local boundedness and continuity of solutions
of the heat equation in the context of Dirichlet spaces under some rather weak
additional assumptions. We consider symmetric local regular Dirichlet forms
which satisfy mild assumptions concerning (a) the existence of cut-off
functions, (b) a local ultracontractivity hypothesis, and (c) a weak
off-diagonal upper bound. In this setting, local weak solutions of the heat
equation, and their time derivatives, are shown to be locally bounded; they are
further locally continuous, if the semigroup admits a locally continuous
density function. Applications of the results are provided including discussion
on the existence of locally bounded heat kernel; $L^\infty$ structure results
for ancient solutions of the heat equation. The last section presents a special
case where the $L^\infty$ off-diagonal upper bound follows from the
ultracontractivity property of the semigroup. This paper is a continuation of
[7].
