Sobolev embeddings into BMO, VMO, and$L$_{∞}

Andrea Cianchi Istituto di Matematica Facoltà di Architettura, Università di Firenze Luboš Pick Mathematical Institute of the Czech Academy of Sciences, Žitná 25, Praha 1, Czech Republic

TBD mathscidoc:1701.332899

Arkiv for Matematik, 36, (2), 317-340, 1997.8
Let$X$be a rearrangement-invariant Banach function space on$R$^{$n$}and let$V$^{1}$X$be the Sobolev space of functions whose gradient belongs to$X$. We give necessary and sufficient conditions on$X$under which$V$^{1}$X$is continuously embedded into BMO or into$L$_{∞}. In particular, we show that$L$_{$n, ∞$}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into BMO and, similarly,$L$_{$n$, 1}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into$L$_{∞}. We further show that$V$^{1}$X$is a subset of VMO if and only if every function from$X$has an absolutely continuous norm in$L$_{$n, ∞$}. A compact inclusion of$V$^{1}$X$into$C$^{0}is characterized as well.
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@inproceedings{andrea1997sobolev,
  title={Sobolev embeddings into BMO, VMO, and$L$_{∞}},
  author={Andrea Cianchi, and Luboš Pick},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203553957581708},
  booktitle={Arkiv for Matematik},
  volume={36},
  number={2},
  pages={317-340},
  year={1997},
}
Andrea Cianchi, and Luboš Pick. Sobolev embeddings into BMO, VMO, and$L$_{∞}. 1997. Vol. 36. In Arkiv for Matematik. pp.317-340. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203553957581708.
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