Compactness of higher-order Sobolev embeddings

Autores: Lenka Slavíková
Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 59, Nº 2, 2015, págs. 373-448
Idioma: inglés
DOI: 10.5565/PUBLMAT_59215_06
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Resumen
- We study higher-order compact Sobolev embeddings on a domain ∩ ⊆Rn endowed with a probability measure ∨ and satisfying certain isoperimetric inequality. Given m ∈ N, we present a condition on a pair of rearrangement-invariant spaces X( ∩,∨) and Y ( ∩,∨) which suffices to guarantee a compact embedding of the Sobolev space V m X ( ∩,∨) into Y (∩,∨). The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of ( ∩,∨). We then apply this result to the characterization of higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, whose standard example is the Gauss space.