Compactness of higher-order Sobolev embeddings

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 $V^{m}X(\Omega ,\nu )$ 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.

References in corpus (1)

• Pointwise Symmetrization Inequalities for Sobolev functions and applications