Sobolev type embeddings in the limiting case. (English) Zbl 0930.46027

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) and let \(k\in\mathbb{N}\). The limiting embedding \[ W^k_p(\Omega)\subset L_\Phi(\Omega,\mu),\quad p={n\over k}, \] in some Orlicz spaces \(L_\Phi\), where \(\mu\) is a suitable measure, attracted a lot of attention since the late sixties. The paper contributes to this subject, generalizing the source space by \(W^k_A\), where \(A\) is a rearrangement-invariant space near the above space \(L_p\), and the optimal target spaces are described in terms of rearrangement, extending and sharpening what is known so far.
Reviewer: H.Triebel (Jena)


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.)
46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological vector spaces
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