Sobolev embeddings into BMO, VMO, and L∞

Abstract

Let X be a rearrangement-invariant Banach function space on Rn and let V1 X be the Sobolev space of functions whose gradient belongs to X. We give necessary and sufficient conditions on X under which V1 X is continuously embedded into BMO or into L∞. In particular, we show that Ln,∞ is the largest rearrangement-invariant space X such that V1 X is continuously embedded into BMO and, similarly, Ln,1 is the largest rearrangement-invariant space X such that V1 X is continuously embedded into L∞. We further show that V1 X is a subset of VMO if and only if every function from X has an absolutely continuous norm in Ln,∞. A compact inclusion of V1 X into C0 is characterized as well.