## 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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### References:

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