Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 625–646

DIRICHLET FORMS AND CONVERGENCE OF BESOV NORMS ON SELF-SIMILAR SETS

Qingsong Gu and Ka-Sing Lau

The Chinese University of Hong Kong, Department of Mathematics, Hong Kong
and Memorial University of Newfoundland, Department of Mathematics and Statistics
NL A1C 5S7, Canada; qsgu 'at' math.cuhk.edu.hk

The Chinese University of Hong Kong, Department of Mathematics, Hong Kong
and University of Pittsburgh, Department of Mathematics
Pittsburgh, Pa. 15217, U.S.A.; kslau 'at' math.cuhk.edu.hk

Abstract. Let B^σ_2,∞, B^σ_2,2 denote the Besov spaces defined on a compact set K ⊂ R^d that is equipped with an α-regular measure μ (K is called an α-set). The critical exponent σ^* is the supremum of the σ such that B^σ_2,2 is dense in C(K). It is known that B^σ_2,2 is the domain of a non-local regular Dirichlet form, and for certain standard self-similar set, B^σ_2,∞ is the domain of a local regular Dirichlet form. In this paper, we study, on the homogenous p.c.f. self-similar sets (which are α-sets), the convergence of the B^σ_2,2-norm to the B^σ_2,∞-norm as σ ↗ σ^* and the associated Dirichlet forms. The theorem extends a celebrate result of Bourgain, Brezis and Mironescu [4] on Euclidean domains, and the more recent results on some self-similar sets [10, 22, 29].

2010 Mathematics Subject Classification: Primary 28A80; Secondary 46E30, 46E35.

Key words: Besov space, Dirichlet form, Γ-convergence, Harnack inequality, heat kernel, p.c.f. fractal, resistance network and trace.

Reference to this article: Q. Gu and K.-S. Lau: Dirichlet forms and convergence of Besov norms on self-similar sets. Ann. Acad. Sci. Fenn. Math. 45 (2020), 625–646.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4536