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

Abstract

We investigate heat kernel-based and other $p$-energy norms (1<p<\infty) on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blowups. With the weak-monotonicity properties for these norms, we generalise the celebrated Bourgain-Brezis-Mironescu (BBM) type characterization for p\neq2. When there admits a heat kernel satisfying the two-sided estimates, we establish the equivalence of various $p$-energy norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when p=2 (in 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 blowups, including the BBM type characterization and Gagliardo-Nirenberg inequality.