Cocompact embedding theorem for functions of bounded variation into Lorentz spaces

Abstract

We show that the embedding $\dot{\mathrm{BV}}({\mathbb{R}}^N)\hookrightarrow L^{1^{\ast },q}({\mathbb{R}}^N)$, $q>1$ is cocompact with respect to the group and the profile decomposition for $\dot{\mathrm{BV}}({\mathbb{R}}^N)$. This paper extends the cocompactness and profile decomposition for the critical space $L^{1^{\ast }}({\mathbb{R}}^N)$ to Lorentz spaces $L^{1^{\ast },q}({\mathbb{R}}^N)$, $q>1$. A counterexample for $\dot{\mathrm{BV}}({\mathbb{R}}^N)\hookrightarrow L^{1^{\ast },1}({\mathbb{R}}^N)$ not cocompact is given in the last section.