Representations for transformations continuous in the $\textrm {BV}$ norm

Abstract:

Riemann and Lebesgue-type integrations can be employed to represent operators on normed function spaces whose norms are not stronger than sup-norm by $T(f) = \smallint f d\mu$ where $\mu$ is determined by the action of T on the simple functions. The real-valued absolutely continuous functions on [0, 1] are not in the closure of the simple functions in the BV norm, and hence such an integral representation of an operator is not obtainable. In this paper the authors develop a v-integral whose structure depends on fundamental functions different than simple functions. This integral is as computable as the Riemann integral. By using these fundamental functions, the authors are able to obtain a direct, analytic representation of the linear functionals on AC which are continuous in the BV norm in terms of the v-integral. Further, the v-integral gives a characterization of the dual of AC in terms of the space of fundamentally bounded set functions which are convex with respect to length. This space is isometrically isomorphically identified with the space of Lipschitz functions anchored at zero with the norm given by the Lipschitz constant, which in turn is isometrically isomorphic to ${L^\infty }$. Hence a natural identification exists between the classical representation and the one given in this paper. The results are extended to the vector setting.