On the Banach structure of multivariate BV spaces
Volume 548 / 2020
Abstract
We introduce and study multivariate generalizations of the classical $\mathit{BV}$ spaces of Jordan, F. Riesz and Wiener. The family of the spaces introduced contains or is intimately related to important function spaces of modern analysis including $\mathit{BMO}$, $\mathit{BV}$, Morrey spaces and Sobolev spaces of arbitrary smoothness, Besov and Triebel–Lizorkin spaces. We prove under mild restrictions that the $\mathit{BV}$ 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 subspace of the $\mathit{BV}$ space generated by $C^\infty$ functions. As a corollary we obtain the “two stars theorem” asserting that the second dual of this subspace is isometrically isomorphic to the $\mathit{BV}$ space. Our proofs essentially depend on approximation properties of $\mathit{BV}$ spaces, in particular the weak$^*$ density of $C^\infty$ functions. The results presented imply similar ones (known and new) for the above mentioned classical function spaces, which are now obtained by a unified approach.
