Almost-compact embeddings
type: Item
instance of: scholarly article
title: Almost-compact embeddings
zbMATH Open document ID: 1256.46016
author: Lenka Slavíková
published in: Mathematische Nachrichten
publication date: 2012-09-06T00:00:00Z
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6078254
zbMATH Keywords: almost-compact embedding
zbMATH Keywords: Banach function space
zbMATH Keywords: rearrangement-invariant space
zbMATH Keywords: product operator
zbMATH Keywords: almost-everywhere convergence
zbMATH Keywords: Lorentz endpoint space
zbMATH Keywords: Marcinkiewicz endpoint space
MaRDI profile type: Publication
full work available at URL: mana.201100286
OpenAlex ID: W2093676626
cites work: Q4039801
cites work: Absolutely continuous embeddings of rearrangement-invariant spaces
cites work: Compactness of Sobolev imbeddings involving rearrangement-invariant norms
cites work: Compactness of integral operators in Banach function spaces
DOI: 10.1002/MANA.201100286
review text: 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.
reviewed by: Leszek Skrzypczak
description: scientific article; zbMATH DE number 6078254