  We focus on heat kernel-based p-energy norms (1<p<\infty) on bounded and
unbounded metric measure spaces, in particular, weak-monotonicity properties
for different types of energies. Such properties are key to related studies,
under which we generalise the convergence result of Bourgain-Brezis-Mironescu
(BBM) for p\neq2. We establish the equivalence of various p-energy norms and
weak-monotonicity properties when there admits a heat kernel satisfying the
two-sided estimates. Using these equivalences, we verify various
weak-monotonicity properties on nested fractals and their blowups. Immediate
consequences are that, many classical results on p-energy norms hold for such
bounded and unbounded fractals, including the BBM convergence and
Gagliardo-Nirenberg inequality.
