Different degrees of non-compactness for optimal Sobolev embeddings

Abstrakt

The structure of non-compactness of optimal Sobolev embed-dings of m-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bern-stein numbers of such embeddings are obtained.

It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular. (c) 2023 Elsevier Inc. All rights reserved.

Klíčová slova

• Sobolev spaces
• Compactness
• Bernstein numbers
• Singular operators