Multivariate bounded variation functions of Jordan–Wiener type

Abstract

We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers. However, each of the proposed concepts preserves only few properties of Jordan variation which are designed to a selected application. In contrast, the multivariate generalization of the Jordan space presented in this paper preserves all known and reveals some previously unknown properties of the space. These, in turn, are special cases of the basic properties of the introduced spaces proved in the paper. Specifically, the first part of the paper describes structure properties of functions of bounded (k,p)-variation (Vpk functions). It includes assertions on discontinuity sets and pointwise differentiability of Vpk functions and their Luzin type and C∞ approximations. The second part presents results on Banach structure of Vpk spaces, namely, atomic decomposition and constructive characterization of their predual spaces. As a result, we obtain the so-called two-stars theorems describing Vpk spaces as second duals of their separable subspaces consisting of functions of “vanishing variation”.