Maximal non-compactness of Sobolev embeddings

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00544893" target="_blank" >RIV/67985840:_____/21:00544893 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/21:10441254

Výsledek na webu

<a href="https://doi.org/10.1007/s12220-020-00522-y" target="_blank" >https://doi.org/10.1007/s12220-020-00522-y</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s12220-020-00522-y" target="_blank" >10.1007/s12220-020-00522-y</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Maximal non-compactness of Sobolev embeddings

Popis výsledku v původním jazyce

It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces Lp,∞ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as p< ∞. Finally, we show that if the target space is L∞ (which formally is also a weak Lebesgue space with p= ∞), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.

Název v anglickém jazyce

Maximal non-compactness of Sobolev embeddings

Popis výsledku anglicky

It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces Lp,∞ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as p< ∞. Finally, we show that if the target space is L∞ (which formally is also a weak Lebesgue space with p= ∞), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA18-00580S" target="_blank" >GA18-00580S: Prostory funkcí a aproximace</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Geometric Analysis

ISSN

1050-6926

e-ISSN

1559-002X

Svazek periodika

31

Číslo periodika v rámci svazku

9

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

26

Strana od-do

9406-9431

Kód UT WoS článku

000577070400001

EID výsledku v databázi Scopus

2-s2.0-85092492549