Almost-compact embeddings
Let \( X\) and \(Y\) be Banach function spaces over a totally \(\sigma\)-finite measure space \((R,\mu)\). We say that \(X\) is almost-compactly embedded into \(Y\) (we write \(X\overset{*}{\hookrightarrow}Y\)) if \(\lim_{n\rightarrow \infty} \sup\{\|f\chi_{E_n}\|_Y: \|f\|_X\leq 1\}=0\) for any sequence of sets \(E_n\rightarrow \emptyset\) \(\mu\)-a.e. Thus functions from the unit ball in \(X\) have uniformly absolutely continuous norms in \(Y\). The aim of this interesting paper is to establish several general results concerning almost-compact embeddings. In particular, it is proved that the embedding is almost-compact if and only if every bounded sequence in \(X\) that converges a.e. to zero converges strongly to zero in \(Y\). This can be used to prove compactness of the embeddings of Sobolev spaces built upon \(X\) to some another Banach space \(Y\) defined on a nonempty open subset of \({\mathbb R}^d\).NEWLINENEWLINEMoreover, it is proved that if the measure space is completely atomic then the almost-compactness of the embedding is equivalent to its compactness. On the other hand if \((R,\mu)\) is nonatomic of finite measure then \(L^\infty\overset{*}{\hookrightarrow}X\) is equivalent to \(X\not= L^\infty\) and that \(X\overset{*}{\hookrightarrow}L^1\) is equivalent to \(X\not= L^1\).NEWLINENEWLINESome product operators as well as embeddings of Lorentz and Marcinkiewicz spaces are also studied. A characterization of almost-compact embeddings among Lorentz and Marcinkiewicz endpoint spaces is given.
scientific article; zbMATH DE number 6078254