Schur property for jump parts of gradient measures

TL;DR

This paper proves that for weakly null sequences in BV space, the jump parts of their gradients vanish strongly, linking the Dunford–Pettis property of SBV and W^{1,1} spaces.

Contribution

It establishes a new connection between the jump parts of gradients in BV functions and the Dunford–Pettis property in related function spaces.

Findings

Jump parts of gradients tend to zero strongly for weakly null sequences.

Dunford–Pettis property for SBV is equivalent to that for W^{1,1}.

Provides insight into the structure of BV and SBV spaces.

Abstract

We consider weakly null sequences in the Banach space of functions of bounded variation $\mathrm{BV}(\mathbb{R}^d)$. We prove that for any such sequence $\{f_n\}$ the jump parts of the gradients of functions $f_n$ tend to $0$ strongly as measures. It implies that Dunford--Pettis property for the space $\mathrm{SBV}$ is equivalent to the Dunford--Pettis property for the Sobolev space $W^{1,1}.$