Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Cianchi, Andrea; Pick, Luboš; Slavíková, Lenka

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 105-144.

On considère des immersions de Sobolev d'ordre quelconque dans des espaces de fonctions sur des domaines de ${\mathbb{R}}^n$ 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 générale basée sur la réduction des immersions en dimension n à 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 permet de détecter l'espace cible optimal dans les immersions de Sobolev, pour une large famille de normes dans des cas où les techniques habituelles ne s'appliquent pas. En particulier on déduit des nouvelles immersions avec espaces-cibles augmentés même dans le cas d'espace de Sobolev standard.

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of ${\mathbb{R}}^n$ endowed with measures whose decay on balls is dominated by a power d of their radius. Norms in arbitrary rearrangement-invariant 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.

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