Schur property for jump parts of gradient measures

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 $W^{1,1}.$