Dirichlet forms and critical exponents on fractals
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- by Qingsong Gu and Ka-Sing Lau PDF
- Trans. Amer. Math. Soc. 373 (2020), 1619-1652 Request permission
Abstract:
Let $B^{\sigma }_{2, \infty }$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $\alpha$-regular measure $\mu$. The critical exponent $\sigma ^*$ is the supremum of the $\sigma$ such that $B^{\sigma }_{2, \infty } \cap C(K)$ is dense in $C(K)$. It is well known that for many standard self-similar sets $K$, $B^{\sigma ^*}_{2, \infty }$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on the p.c.f. (post critically finite) sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied.References
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Additional Information
- Qingsong Gu
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China
- MR Author ID: 1308143
- Email: qsgu@math.cuhk.edu.hk
- Ka-Sing Lau
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China; and School of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China; and Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15217
- MR Author ID: 190087
- Email: kslau@math.cuhk.edu.hk
- Received by editor(s): April 20, 2018
- Received by editor(s) in revised form: April 4, 2019
- Published electronically: December 17, 2019
- Additional Notes: This research was supported in part by the HKRGC grant.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 1619-1652
- MSC (2010): Primary 28A80; Secondary 46E30, 46E35
- DOI: https://doi.org/10.1090/tran/8004
- MathSciNet review: 4068276