Absolutely continuous functions on compact and connected 1-dimensional metric spaces

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 1Ahlfors 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.