Cocompact imbedding theorem for functions of bounded variation into Lorentz spaces

TL;DR

This paper proves cocompact embedding and profile decomposition for functions of bounded variation into Lorentz spaces, extending known results from Lebesgue spaces and providing a counterexample for the limiting case.

Contribution

It extends cocompactness and profile decomposition results from critical Lebesgue spaces to Lorentz spaces for BV functions, and identifies the non-cocompact case.

Findings

Cocompact embedding of BV into L^{1*,q} for q>1.

Profile decomposition for BV in Lorentz spaces.

Counterexample showing non-cocompactness for q=1.

Abstract

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