Absolutely continuous embeddings of rearrangement-invariant spaces
This paper is devoted to the study of absolutely continuous embeddings between rearrangement invariant spaces, a property closely related to compactness (see, for instance, \textit{E.~Pustylnik} [``On compactness of Sobolev embeddings in rearrangement-invariant spaces'', Forum Math. 18, No.~5, 839--852 (2006; Zbl 1120.46019)]). Embeddings between rearrangement invariant spaces are never compact and absolute continuity is a suitable substitute of compactness that improves our knowledge on relations among rearrangement invariant spaces. An embedding between rearrangement invariant spaces \(Y \hookrightarrow X\) is said to be absolutely continuous if \[ \forall \varepsilon >0, \; \exists \delta >0 \; \text{ such that } \|f \chi_{D}\|_{X} < \varepsilon\;\text{ for all } f\in B_{Y}, \] if the measure of \(D\) is less than \(\delta\). As a brief description of the results the paper presents, it must be mentioned that: {\parindent6.5mm \begin{itemize}\item[(1)] It is proved that absolutely continuous embeddings between rearrangement invariant spaces are never optimal. \item[(2)] Necessary conditions for the embedding \(Y\hookrightarrow X\) to be absolutely continuous are given in terms of the fundamental functions of the spaces \(X\) and \(Y\). \item[(3)] Sufficient conditions for absolute continuity of the embedding \(Y\hookrightarrow X\) are also shown. \item[(4)] In Section~4, the authors define the function \(\sigma_{Y,X}(h)\), \(h>0\), that generalizes the one introduced by \textit{J.~Appell} and \textit{E.~De Pascale} [``Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili'', Boll. Boll. Unione Mat. Ital., VI. Ser., B 3, 497--515 (1984; Zbl 0507.46025)]. \(\sigma_{Y,X}\) is used to give quantitative estimates of the degree of absolute continuity of the embedding \(Y\hookrightarrow X\) by the magnitude of the infimum \(\sigma_{Y,X}(h)\) when \(h \to 0\). \end{itemize}} If \(\lim_{h \to 0} \sigma_{Y,X}(h) \not =0\), then the embedding \(Y \hookrightarrow X\) is not absolutely continuous.
scientific article; zbMATH DE number 5841054