TY - JOUR
T1 - Besov class via heat semigroup on Dirichlet spaces III
T2 - BV functions and sub-Gaussian heat kernel estimates
AU - Alonso-Ruiz, Patricia
AU - Baudoin, Fabrice
AU - Chen, Li
AU - Rogers, Luke
AU - Shanmugalingam, Nageswari
AU - Teplyaev, Alexander
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2021/10
Y1 - 2021/10
N2 - 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 L1 Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in L1. The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.
AB - 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 L1 Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in L1. The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.
UR - http://www.scopus.com/inward/record.url?scp=85111252534&partnerID=8YFLogxK
U2 - 10.1007/s00526-021-02041-2
DO - 10.1007/s00526-021-02041-2
M3 - Journal article
AN - SCOPUS:85111252534
SN - 0944-2669
VL - 60
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 5
M1 - 170
ER -