A Characterization of Compact Parts of Lp Spaces Application to Sobolev Embeddings

Abstract

To characterize compact parts of spaces $L^p(\Omega)$, we introduce a concept of equi-integrability based on the approximation of elements of $L^p(\Omega)$ by simple functions. The resulting theorem will be used to develop a new methodology to prove and extend results about the compactness of Sobolev embeddings.