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We show that the imbedding $\dot{BV}(\mathbb{R}^N)\hookrightarrowL^{1^\ast,q}(\mathbb{R}^N)$, $q>1$ is cocompact with respect to group and theprofile decomposition for $\dot{BV}(\mathbb{R}^N)$. This paper extends thecocompactness 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 $\dot{BV}(\mathbb{R}^N)\hookrightarrowL^{1^\ast,1}(\mathbb{R}^N)$ not cocompact is given in the last section.

Cocompact imbedding theorem for functions of bounded variation into Lorentz spaces