Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces

Abstract

We review some of the recent developments in the theory of Korevaar-Schoen-Sobolev spaces on metric measure spaces and also present some new results. While this theory coincides with those of Cheeger and Shanmugalingam if the space is doubling and satisfies a Poincaré inequality, it offers new perspectives in the context of fractals for which the approach by weak upper gradients is inadequate.

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

“…In all these studies, certain energy-control conditions (which will be called weak-monotonicity properties in this paper) are essentially required and important, to guarantee 'some level of L p infinitesimal regularity and global controlled L p geometry' as [7] stated. Relating different energycontrol conditions is a main contribution of our paper.…”

“…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.…”

“…By the time we were about to finish our paper, we notice a lecture note [7] from Baudoin, where we find out some very recent work related to the BBM convergence with rather different viewpoint (see [14,22]), and the verification of properties (KE) and (NE) for two nested fractals using a different argument (see [1,7] ). We examine properties (KE) and (NE) by their equivalence to property (VE), and this equivalence does not rely on the finitely-ramified property.…”

“…property. Alonso Ruiz, Baudoin et al established two analogues of classical analysis results under property (KE) or (NE) for 1 < p < ∞, that is, the Sobolev-type inequality in [2, Theorem 1.1] and the full scale of Gagliardo-Nirenberg inequality introduced by [7,Theorem 4.3 and Theorem 4.4]. The latter inequality only holds for unbounded spaces in [7], and this motivates us to study unbounded fractals.…”

“…Whenever the critical exponents appear in our paper, we assume that they are finite, and it is known by [7,Theorem 4.2] that when the chain condition holds, then…”