Compactness of higher-order Sobolev embeddings

TL;DR

This paper establishes conditions for the compactness of higher-order Sobolev embeddings on various measure spaces, linking geometric properties with functional analysis to generalize classical results.

Contribution

It provides a new criterion based on isoperimetric inequalities and rearrangement-invariant spaces for ensuring compact Sobolev embeddings in diverse settings.

Findings

Derived a condition involving one-dimensional operators and isoperimetric functions.

Applied the criterion to John domains, Maz'ya classes, and Gaussian spaces.

Extended classical Sobolev embedding compactness results to more general measure spaces.

Abstract

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.