Abstract
LetX be a rearrangement-invariant Banach function space onR n and letV 1 X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV 1 X is continuously embedded into BMO or intoL ∞. In particular, we show thatL n, ∞ is the largest rearrangement-invariant spaceX such thatV 1 X is continuously embedded into BMO and, similarly,L n, 1 is the largest rearrangement-invariant spaceX such thatV 1 X is continuously embedded intoL ∞. We further show thatV 1 X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inL n, ∞ . A compact inclusion ofV 1 X intoC 0 is characterized as well.
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Cianchi, A., Pick, L. Sobolev embeddings into BMO, VMO, andL ∞ . Ark. Mat. 36, 317–340 (1998). https://doi.org/10.1007/BF02384772
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DOI: https://doi.org/10.1007/BF02384772