We give necessary and sufficient conditions for sub-Gaussian estimates of
the heat kernel of a strongly local regular Dirichlet form on a metric measure space.
The conditions for two-sided estimates are given in terms of the generalized capacity
inequality and the Poincar´e inequality. The main difficulty lies in obtaining the elliptic
Harnack inequality under these assumptions. The conditions for upper bound alone are
given in terms of the generalized capacity inequality and the Faber-Krahn inequality.