Symmetry and compact embeddings for critical exponents in metric-measure spaces

TL;DR

This paper establishes a compact Sobolev embedding for H-invariant functions in compact metric-measure spaces, extending classical results to more general symmetry groups and critical exponents.

Contribution

It introduces a new compact embedding result for invariant functions under measure-preserving group actions in metric-measure spaces, generalizing known isometry-based embeddings.

Findings

Proves compact Sobolev embeddings for H-invariant functions

Extends classical isometry-invariant embeddings to broader symmetry groups

Applicable to Riemannian manifolds with volume-preserving diffeomorphisms

Abstract

We obtain a compact Sobolev embedding for $H$-invariant functions in compact metric-measure spaces, where $H$ is a subgroup of the measure preserving bijections. In Riemannian manifolds, $H$ is a subgroup of the volume preserving diffeomorphisms: a compact embedding for the critical exponents follows. The results can be viewed as an extension of Sobolev embeddings of functions invariant under isometries in compact manifolds.