Open Access
2015 Compactness of higher-order Sobolev embeddings
Lenka Slavíková
Publ. Mat. 59(2): 373-448 (2015).


We study higher-order compact Sobolev embeddings on a domain $\Omega \subseteq \mathbb R^n$ endowed with a probability measure $\nu$ and satisfying certain isoperimetric inequality. Given $m\in \mathbb N$, we present a condition on a pair of rearrangement-invariant spaces $X(\Omega,\nu)$ and $Y(\Omega,\nu)$ which suffices to guarantee a compact embedding of the Sobolev space $V^mX(\Omega,\nu)$ into $Y(\Omega,\nu)$. The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of $(\Omega,\nu)$. 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.


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Lenka Slavíková. "Compactness of higher-order Sobolev embeddings." Publ. Mat. 59 (2) 373 - 448, 2015.


Published: 2015
First available in Project Euclid: 30 July 2015

zbMATH: 1345.46032
MathSciNet: MR3374613

Primary: 46E30 , 46E35

Keywords: almost-compact embedding , compactness , ‎integral operator , isoperimetric function , John domain , Maz'ya domain , product probability space , rearrangement-invariant space , Sobolev space

Rights: Copyright © 2015 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.59 • No. 2 • 2015
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