English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Some necessary and sufficient conditions on rearrangement-invariant spaces  A, B  are stated, which provide compactness of the corresponding Sobolev embedding  W   A    k  (Ω) ↪  B (Ω) for any bounded open set Ω ⊂ ℝ  n  . It is shown that this property is equivalent to compactness of some integral operator of Hardy type acting from  A  to  B . This enables the compactness problem to be connected with behaviour of the ratio of fundamental functions ϕ  A  ( t )/ϕ  B  ( t ) at zero. We also show that any optimal Sobolev embedding is not compact and extend this result to some other Sobolev embeddings “close” to optimal one.

On compactness of Sobolev embeddings in rearrangement-invariant spaces