On maps of bounded $p$-variation with $p>1$

Abstract

This paper addresses properties of maps of bounded p-variation (p>1) in the sense of N. Wiener, which are defined on a subset of the real line and take values in metric or normed spaces. We prove the structural theorem for these maps and study their continuity properties. We obtain the existence of a Holder continuous path of minimal p-variation between two points and establish the compactness theorem relative to the p-variation, which is an analog of the well-known Helly selection principle in the theory of functions of bounded variation. We prove that the space of maps of bounded p-variation with values in a Banach space is also a Banach space. We give an example of a Holder continuous of exponent 0<γ<1 set-valued map with no continuous selection. In the case p=1 we show that a compact absolutely continuous set-valued map from the compact interval into subsets of a Banach space admits an absolutely continuous selection.