Isometries on spaces of absolutely continuous vector-valued functions

Abstract

This paper investigates the surjective (not necessarily linear) isometries between spaces of absolutely continuous vector-valued functions with respect to the norm ‖⋅‖=max⁡{‖⋅‖∞,V(⋅)}, where ‖⋅‖∞ and V(⋅) denote the supremum norm and the total variation of a function, respectively, and gives an absolutely continuous version of a celebrated theorem by Jerison. As a consequence, in the scalar-valued case, we obtain generalizations of all known results concerning such isometries by a different approach.