arXiv subjects

Carlo Sbordone

Publications and source records attributed to Carlo Sbordone.

3 recordsLinked to original sources

arXiv2020

Preduals and double preduals of some Banach spaces

In this paper we study ways to establish when a Banach space can be identified as the dual or the double dual of another Banach space. To obtain these results, we relate these spaces with other, concrete Banach spaces - tipically $\ell^1$ and $\ell^\infty$ - and show that under suitable assumptions we can transfer properties of these spaces to the space we consider. In particular, we show how these results can be used to obtain in a simple way interesting results about spaces such as $BMO$ and $BV$.

math.FA↗

arXiv2020

On BV homeomorphisms

We obtain the rectifiability of the graph of a bounded variation homeomorphism $f$ in the plane and relations between gradients of $f$ and its inverse. Further, we show an example of a bounded variation homeomorphism $f$ in the plane which satisfies the $(N)$ and $(N^{-1})$ properties and strict positivity of Jacobian of both itself and its inverse, but neither $f$ nor $f^{-1}$ is Sobolev.

math.CA↗

arXiv2019

Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces

Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\mathcal{B}_\ast$, the biduality result that $\mathcal{B}_0^\ast = \mathcal{B}_\ast$ and $\mathcal{B}_\ast^\ast = \mathcal{B}$, and a formula for the distance from an element $f \in \mathcal{B}$ to $\mathcal{B}_0$.

math.FA↗