Sobolev embeddings into BMO, VMO, andL∞
LetX be a rearrangement-invariant Banach function space onRn and letV1X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV1X is continuously embedded into BMO or intoL∞. In particular, we show thatLn, ∞ is the largest rearrangement-invariant spaceX such thatV1X is continuously embedded into BMO and, similarly,Ln, 1 is the largest rearrangement-invariant spaceX such thatV1X is continuously embedded intoL∞. We further show thatV1X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inLn, ∞. A compact inclusion ofV1X intoC0 is characterized as well.