Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

Abstract

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global L 1 Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in L 1 . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

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

“…We start with a short review of some properties of the Besov-Lipschitz spaces that will be useful in the sequel. The theory of Besov classes on doubling metric measure spaces is rich and the literature on this topic is nowadays quite large so we will not try to be exhaustive; For references related to the discussion below see for instance [1], [5,3,4], [20] , [24] and [52].…”

“…Proof. The case p = 1 was first treated in [4,Theorem 5.1] where it is actually proved that for any nested fractal α 1 = d h and that the corresponding Korevaar-Schoen-Sobolev space is always dense into L 1 . The case 1 < p ≤ 2 was then treated for the first time in [1,Theorem 3.10] where it is proved that in that case α p = 1 + d h −1 p .…”

“…Furthermore, an appealing aspect of the theory of Korevaar-Schoen-Sobolev spaces is its close connection to the very rich theory of heat kernels and Dirichlet forms as was developed in [5,3,4] after [37], [20], [42], and [46]. Due to this connection, one can hope to export to a general metric measure space setting some of the powerful heat kernel techniques, like the Bakry-Émery-Ledoux machinery, see [3,4].…”

“…We show that those two examples satisfy for every p ≥ 1 the inequality (1) for some value α = α p and as a consequence prove new Nash inequalities. Finally, in Section 7, we review some of the results in [5,3,4] about the connection between the Dirichlet forms theory and the Korevaar-Schoen-Sobolev spaces. Throughout the text several open problems and possible research directions are discussed.…”

Fractal polynomials On the Sierpiński gasket and some dimensional results

“…One is demonstrated by Verma et al in [30], Remark (4.13) . Recall that every non-constant harmonic function h is not of bounded variation , and the other one is proposed by Ruiz et al in [4], Theorem 5.2 . Recall that on the SG any non-constant piecewise harmonic function is not in bounded variation.…”