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## Studia Mathematica

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## Compactness of Sobolev embeddings and decay of norms

### Tom 265 / 2022

Studia Mathematica 265 (2022), 1-35 MSC: Primary 46E30; Secondary 46E35. DOI: 10.4064/sm201119-29-9 Opublikowany online: 24 February 2022

#### Streszczenie

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.

#### Autorzy

• Jan LangDepartment of Mathematics
The Ohio State University
231 West 18th Avenue
Columbus, OH 43210-1174, USA
e-mail
• Zdeněk MihulaDepartment of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
Sokolovská 83
186 75 Praha 8, Czech Republic
and
Department of Mathematics
Faculty of Electric Engineering
Czech Technical University in Prague
Technická 2
166 27 Praha 6, Czech Republic
e-mail
e-mail
• Luboš PickDepartment of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
Sokolovská 83
186 75 Praha 8, Czech Republic
e-mail

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