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On sharp higher order Sobolev embeddings. (English) Zbl 1108.46029

It is well-known that the classical Sobolev embedding theorem
\[ W_0^{k,p}(\Omega) \subset L^q(\Omega),\;{1\over q} = {1\over p} - {k \over n},\;1< p < {n \over k}, \tag{1} \]
fails in the limit case \(p = {n \over k}\). The main result of the present paper is an extension of (1) to this case, by replacing \(L^\infty(\Omega)\) with a more exotic space denoted by \(L(\infty, p)(\Omega)\). If \(1 < p \leq \infty\), \(0 < q \leq \infty\), \(L(p,q)(\Omega)\) stands for the class of measurable functions such that \[ \int_0^\infty \{[(f^{**}(t) - f^*(t)) t^{1/p}]^q\, {dt/t}\}^{1/q} < \infty. \] Here, \(f^*\) is the usual decreasing rearrangement function, while, for \(t > 0\), \[ f^{**}(t) = {1\over t} \int_0^t f^*(u)\, du. \] The result is optimal in the sense that, if \(X(\Omega)\) is a rearrangement invariant space containing \(W_0^{{n \over k},p}(\Omega)\), then \(X(\Omega)\) is contained in \(L(\infty,p)(\Omega)\).

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.)
26D10 Inequalities involving derivatives and differential and integral operators
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