Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation

TL;DR

This paper develops new inequalities in metric spaces related to fractional Sobolev spaces, using symmetrization and isoperimetric profiles, with applications to embeddings, measures, and growth estimates.

Contribution

It introduces novel oscillation inequalities involving the Peetre K-functional and isoperimetric profiles, advancing the understanding of fractional Sobolev inequalities in various metric measure spaces.

Findings

Established new oscillation inequalities in metric spaces.

Analyzed fractional Sobolev and Morrey-Sobolev embeddings in different contexts.

Derived growth estimates for generalized Sobolev and Besov spaces.

Abstract

We obtain new oscillation inequalities in metric spaces in terms of the Peetre $K-$functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular we include a detailed study of Gaussian measures as well as probablity measures between Gaussian and exponential. We show a kind of reverse Polya-Szego principle that allows us to obtain continuity as a self improvement from boundedness, using symetrization inequalities. Our methods also allow for precise estimates of growth envelopes of generalized Sobolev and Besov spaces on metric spaces. We also consider embeddings into $BMO$ and their connection to Sobolev embeddings.