Compactness for Sobolev-type trace operators

Cavaliere, Paola; Mihula, Zdeněk

doi:10.1016/j.na.2019.01.013

Cited by 8 publications

(6 citation statements)

References 13 publications

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“…Of particular interest is an embedding into a Lorentz space whose second index is equal to infinity, equivalent to a weak Lebesgue space, see [8]. However, it is a rule of thumb that if a Sobolev embedding is sharp in the sense of function spaces, then it is never compact [9,19,29,30]. It is thus of interest to study how bad is its non-compactness.…”

Section: Examplementioning

confidence: 99%

Maximal non-compactness of Sobolev embeddings

Lang,

Musil,

Olšák

et al. 2020

Preprint

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It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are noncompact 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 L p,∞ 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 noncompactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.2000 Mathematics Subject Classification. 41A46, 46E35.

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Section: Examplementioning

confidence: 99%

Maximal non-compactness of Sobolev embeddings

Lang,

Musil,

Olšák

et al. 2020

Preprint

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Section: Introductionmentioning

confidence: 99%

“…Reduction principles for Sobolev type inequalities, for different kinds of norms or domains, are the subject of [3,29,30,31]. Compactness of Sobolev embeddings is characterized via reduction principles in [10,49].New results. In the present paper we abandon the point of view of linking Sobolev to isoperimetric inequalities, and pursue the approach to a wider family of Sobolev inequalities, via reduction principles, from a different perspective.…”

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confidence: 99%

Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Cianchi

Pick

Slavíková

2020

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of R n endowed with measures whose decay on balls is dominated by a power d of their radius. Norms in arbitrary rearrangementinvariant spaces are contemplated. A comprehensive approach is proposed based on the reduction of the relevant n-dimensional embeddings to one-dimensional Hardy-type inequalities. Interestingly, the latter inequalities depend on the involved measure only through the power d. Our results allow for the detection of the optimal target space in Sobolev embeddings, for broad families of norms, in situations where customary techniques do not apply. In particular, new embeddings, with augmented target spaces, are deduced even for standard Sobolev spaces. RésuméOn considère des immersions de Sobolev d'ordre quelconque dans des espaces de fonctions sur des domaines de R n munis des mesures avec une tendance dans les boules qui est dominée par une puissance d du rayon. Des normes dans les espaces arbitraires invariants par réarrangements sont permises. Nous proposons une approche général basée sur la réduction des immersions en dimension nà des inégalités du type Hardy en dimension un. On souligne que ces inégalités dépendent de la mesure considérée seulement par le degré de puissance d. Notre résultat permets de détecter l'espace cible optimal dans les immersions de Sobolev, pour une large famille de normes dans des cas où le techniques habituelles ne s'appliquent pas. En particulier on déduit des nouvelles immersions avec espaces cible augmentés même dans le cas d'espace de Sobolev standard.Mathematics Subject Classification: 46E35, 46E30.holds for some constant C 2 and every f ∈ X(0, 1). The same characterization applies to any domain Ω with |Ω| < ∞, provided that the space W m X(Ω) is replaced with W m 0 X(Ω). This special case was earlier obtained, via a different approach, in [33]. Reduction principles for Sobolev type inequalities, for different kinds of norms or domains, are the subject of [3,29,30,31]. Compactness of Sobolev embeddings is characterized via reduction principles in [10,49].New results. In the present paper we abandon the point of view of linking Sobolev to isoperimetric inequalities, and pursue the approach to a wider family of Sobolev inequalities, via reduction principles, from a different perspective. A basic version of the inequalities that will be considered, called Sobolev trace inequalities in a broad sense, has the form u Y (Ω,µ) ≤ C ∇ m u X(Ω)

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“…These advances, introduced in [15] and then further developed in many works, see e.g. [26,11], paved the way for studying deeper properties of Sobolev embeddings, a pivotal example of which is compactness, see [27,39,38,8].…”

Section: Introductionmentioning

confidence: 99%

Compactness of Sobolev embeddings and decay of norms

Lang,

Mihula,

Pick

2020

Preprint

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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 its 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.

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Compactness for Sobolev-type trace operators

Cited by 8 publications

References 13 publications

Maximal non-compactness of Sobolev embeddings

Maximal non-compactness of Sobolev embeddings

Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Compactness of Sobolev embeddings and decay of norms

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