Cocompact embedding theorem for functions of bounded variation into Lorentz spaces

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.