spaces'
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How to cite this paper {#how-to-cite-this-paper .unnumbered}
======================

This paper has been accepted for publication in *Journal of Functional Analysis* and is available on

<https://doi.org/10.1016/j.jfa.2019.108341>.

Should you wish to cite this paper, the authors would like to cordially ask you to cite it appropriately.

Introduction {#sec1}
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Given function spaces $X,Y$ and an operator $T$ that maps $X$ boundedly into $Y$, it is natural to ask whether there is a space bigger than $X$ that is also mapped boundedly by $T$ into $Y$, or a space smaller than $Y$ into which $T$ maps $X$ boundedly.

Such questions have been attracting a great deal of attention for many years, in particular in connection with embeddings of Sobolev spaces, see, for example [@BMR:03; @CarSo:97; @ClaSo:16a; @ClaSo:16; @CwPu:98; @FiRa:06; @Ker:79; @MaMi:06; @MMP:06; @MaMi:10; @Tal:94; @Tal:16; @Tar:98]. By way of illustration we consider a particularly simple Sobolev embedding. Let $\Omega $ be a bounded open subset of $\mathbb{R}^{n}$, let $p\in [1,n)$ and put $p^{\ast }=np/(n-p)$. It is classical that, in standard notation, the Sobolev space $W_{0}^{1,p}(\Omega )$ is embedded in $L^{p^{\ast }}(\Omega )$. Can $W_{0}^{1,p}(\Omega )$ be embedded in a space smaller than $L^{p^{\ast }}(\Omega )$? Is there a space larger than 