On Banach Structure of Multivariate BV Spaces I

TL;DR

This paper introduces multivariate BV spaces, explores their duality and atomic decompositions, and connects them with classical function spaces like BMO, Sobolev, and Besov spaces, providing new structural insights.

Contribution

It generalizes classical BV spaces to multivariate settings, establishes duality and atomic decompositions, and links these spaces to well-known modern analysis function spaces.

Findings

BV spaces are dual under mild conditions

Constructive atomic decompositions of preduals

Second dual is isometrically isomorphic to BV space

Abstract

We introduce and study multivariate generalizations of the classical BV spaces of Jordan, F. Riesz and Wiener. The family of the introduced spaces contains or is intimately related to a considerable class of function spaces of modern analysis including BMO, BV, Morrey spaces and those of Sobolev of arbitrary smoothness, Besov and Triebel-Lizorkin spaces. We prove under mild restrictions that the BV spaces of this family are dual and present constructive characterizations of their preduals via atomic decompositions. Moreover, we show that under additional restrictions such a predual space is isometrically isomorphic to the dual space of the separable subspace of the related BV space generated by $C^\infty$ functions. As a corollary we obtain the "two stars theorem" asserting that the second dual of this separable subspace is isometrically isomorphic to the BV space. An essential role in the proofs play approximation properties of the BV spaces under consideration, in particular, weak$^*$ denseness of their subspaces of $C^\infty$ functions. Our results imply the similar ones (old and new) for the classical function spaces listed above obtained by the unified approach.