Heat kernel-based p -energy norms on metric measure spaces

Abstract We investigate heat kernel-based and other p -energy norms ( $1\lt p\lt\infty$ ) on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blow-ups. With the weak-monotonicity properties for these semi-norms, we generalise the celebrated Bourgain–Brezis-Mironescu (BBM) type characterisation for $p\neq2$ . When the underlying space admits a heat kernel satisfying the sub-Gaussian estimates, we establish the equivalence of various p -energy semi-norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when $p=2$ (that is the case of Dirichlet form). Our paper’s key results concern the equivalence and verification of various weak-monotonicity properties on fractals. Consequently, many classical results on p -energy norms hold on nested fractals and their blow-ups, including the BBM type characterisation and Gagliardo-Nirenberg inequality.