Compactness of Sobolev embeddings and decay of norms

Abstract

We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with d-Ahlfors measures under certain restriction on the speed of their decay on balls. We show that the gateway to compactness of such embeddings, while formally describable by means of optimal embeddings and almost-compact embeddings, is quite elusive. It is known that such a Sobolev embedding is not compact when its target space has the optimal fundamental function. We show that, quite surprisingly, such a target space can actually be "fundamentally enlarged", and yet the resulting embedding remains noncompact. In order to do that, we develop two different approaches. One is based on enlarging the optimal target space itself, and the other is based on enlarging the Marcinkiewicz space corresponding to the optimal fundamental function.

Maximal Non-compactness of Sobolev Embeddings

“…Notably, the target space in (1.6) is neither a Marcinkiewicz space nor it is disjointly superadditive (the latter observation, although it is most likely not available in an explicit form, is hidden in [23], as we shall point out), and so the techniques that successfully worked in earlier approaches cannot be used here. As for (1.7), although the target space is a Marcinkiewicz space as in [26], the method used there is not suitable for proving the maximal noncompactness of (1.7). The reason is that the logarithmic function defining the Marcinkiewicz space L ∞,∞, 1 n −1 (Ω) grows too slowly (or rather it is too slowly varying).…”