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 Ln, ∞ is the largest rearrangement-invariant space X such that V1X is continuously embedded into BMO and, similarly, Ln, 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 Ln, ∞. A compact inclusion of V1X into C0 is characterized as well.
Citation
Andrea Cianchi. Luboš Pick. "Sobolev embeddings into BMO, VMO, and L∞." Ark. Mat. 36 (2) 317 - 340, October 1998. https://doi.org/10.1007/BF02384772
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