Sobolev embeddings into BMO, VMO, and L∞

Abstract

LetX be a rearrangement-invariant Banach function space on Rⁿ and let V¹X be the Sobolev space of functions whose gradient belongs to X. We give necessary and sufficient conditions on X under which V¹X is continuously embedded into BMO or into L_∞. In particular, we show that L_{n, ∞} is the largest rearrangement-invariant space X such that V¹X is continuously embedded into BMO and, similarly, L_{n, 1} is the largest rearrangement-invariant space X such that V¹X is continuously embedded into L_∞. We further show that V¹X 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 V¹X into C⁰ is characterized as well.