Convergence of p-Energy Forms on Homogeneous p.c.f Self-Similar Sets

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

“…Historical background and motivation. Our original motivation is generalising the celebrated 'Bourgain-Brezis-Mironescu (BBM) convergence' (1.1) of p-energy forms (1 < p < ∞) in [13] to bounded and unbounded metric measure space, which also extends the classical convergence of non-local Dirichlet forms to a local one on unbounded metric measure spaces or fractals (see [30] and [38]). This originates from Bourgain, Brezis and Mironescu in [8, where D is a smooth area in R n , which states that multiplying by a scaling factor 1 − σ, the fractional Gagliardo semi-norm of a function converges to the first-order Sobolev semi-norm as σ → 1.…”

“…For a smooth Euclidean area D, the penergy is simply defined as D |∇u(x)| p dx (the right-hand side of (1.1)), but for fractals or metric measure spaces, it is not easy to define proper gradient structure to characterise the smoothness of certain 'core' functions arise naturally from analysis. We refer to [13] which considered the case 1 < p < ∞ on homogeneous p.c.f. self-similar sets.…”

“…Heat kernel-based p-energy norms basically appeared in [31], and later heat semigroup-based norms are systematically studied by Alonso Ruiz, Baudoin et al for 1 ≤ p < ∞ in [2,3,4], where they focus on generalising classical analysis results including Sobolev embeddings, isoperimetric inequalities and the class of bounded variation functions etc. It is possible to use heat kernel-based p-energy norms to construct equivalent (or exactly the same) p-energy given by the graph-approximation approach, to satisfy further restrictions like convexity or self-similarity for self-similar sets, see for example [9,13].…”

“…The key to the BBM convergence is that, we need to apply proper weak-monotonicity properties on metric measure spaces (to replace 'property (E)' used for fractals). We apply the concept of property (KE) raised in [2,Definition 6.7] and (NE) raised in [7,Definition 4.5] (all termed P(p, α) therein), and use (VE) for bounded and unbounded fractals based on [13,Definition 3.1], where the letters 'E' stands for energy control and 'K','N','V' stands for (heat) kernel, (Besov) norm, vertex respectively. We also consider their slightly weaker variants by replacing 'sup' with 'limsup' for BBM convergence.…”

“…( In [13], we obtain the convergence of non-local p-energy forms to local p-energy forms using discrete energy-control property (E) (see Definition 4.7). Here we use a more general condition (VE), which also motivates us to explore the equivalence of weak-monotonicity properties (VE) and (KE), (NE) on fractal spaces.…”