Year: 2013
Type: article
Abstract: We study higher-order compact Sobolev embeddings on a domain $Ω\subseteq \mathbb R^n$ endowed with a probability measure $ν$ and satisfying certain isoperimetric inequality. Given $m\in \mathbb N$, we present a condition on a pair of rearrangement-invariant spaces $X(Ω,ν)$ and $Y(Ω,ν)$ which suffices to guarantee a compact embedding of the Sobolev space $V^mX(Ω,ν)$ into $Y(Ω,ν)$. The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of $(Ω,ν)$. 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.
Author: Slavíková, Lenka
Institution: Charles University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1639407239, G7356062917
OpenAlex ID: W2949090296