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Sobolev embeddings into BMO, VMO, and \(L_{\infty}\). (English) Zbl 1035.46502

The main result of this paper is a characterization of all rearrangement invariant Banach function spaces \(X\) such that the associated Sobolev spaces \(S\) of functions whose gradients belong to \(X\) is continuously imbedded in the John Nirenberg class of bounded mean oscillation functions or in the class \(L_\infty\). The Marcinkiewicz space \( L_{n,\infty}\) is the largest rearrangement invariant space \(X\) satisfying the continuous imbedding of \(S\) into BMO. Then it is shown that the Lorentz space \(L_{n,1}\) is the largest rearrangement invariant space \(X\) such that \(S\) is continuously imbedded in the space \(L_\infty\). Finally, the authors show a necessary and sufficient condition for \(S\) to be uniformly included in the Sarason class of vanishing mean oscillation functions.
We wish to point out that basic steps for the imbedding results are two interesting generalizations of the Pólya-Szegö inequality in the context of rearrangement invariant spaces.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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