Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 281-291

ABSOLUTELY CONTINUOUS FUNCTIONS ON COMPACT AND CONNECTED 1-DIMENSIONAL METRIC SPACES

Xiaodan Zhou

Worcester Polytechnic Institute, Mathematical Sciences Department
Worcester, MA 01609-2280, U.S.A.; xzhou3 'at' wpi.edu

Abstract. In this paper, we study the absolutely continuous characterization of Sobolev functions on compact and connected 1-dimensional metric spaces X. We generalize the definition of absolutely continuous functions to such spaces and prove the equivalence between the absolutely continuous functions and Newtonian Sobolev functions. We also show that a compact and 1-Ahlfors regular metric space X supports a p-Poincaré inequality for 1 ≤ p ≤ ∞ if and only if X is quasiconvex. As a result, the absolutely continuous functions are equivalent to the Sobolev functions defined via several different approaches.

2010 Mathematics Subject Classification: Primary 46E35, 54E45.

Key words: Metric spaces, absolute continuity, Poincaré inequality, Sobolev spaces.

Reference to this article: X. Zhou: Absolutely continuous functions on compact and connected 1-dimensional metric spaces. Ann. Acad. Sci. Fenn. Math. 44 (2019), 281-291.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4412