c © 1997 Universitat de Barcelona Extrapolation of Sobolev imbeddings

We survey recent results on limiting imbeddings of Sobolev spaces, particularly, those concerning weakening of assumptions on integrability of derivatives, con-sidering spaces with dominating mixed derivatives and the case of weighted spaces. 1. A quintessence This survey deals with some of the recent results on limiting imbeddings of Sobolev spaces and their generalizations. As is well known the limiting imbedding theorem for Sobolev spaces states ([42], [35]) that Wm,p(Ω), where Ω ⊂ RN is a bounded domain with a sufficiently smooth boundary, 1 < p <∞, and mp = N, is imbedded into an Orlicz space LΦ(Ω) with Φ(t) = exp tN/(N−m) − 1. Since Wm,p↪→Lq(Ω) if mp = N and q is an arbitrary finite positive number the limiting imbedding can be considered as an improvement or a refinement of these imbeddings in the framework of a finer scale of target spaces, namely of Orlicz spaces. We recall that the limiting imbedding is also sharp in terms of usual ordering of Young functions (Φ1 ≺ Φ2 if lim t→∞Φ1(λt)/Φ2(t) = 0 for all λ> 0). Different techniques of proofs have usually a common underlying idea, namely, u ∈ LΦ(Ω), with Φ as above if (1.1) ‖u|Lq ‖ ≤ cq1−m/N 601 602 Krbec for ‖u|Wm,p ‖ ≤ 1, where c is independent of u and q, that is, equivalently, if (1.2) sup 1<q<∞ q−(1−m/N)‖u|Lq ‖ <∞. Similarly, the condition lim p→