Improved Fractional Sobolev Embeddings on Closed Riemannian Manifolds under Isometric Group Actions

TL;DR

This paper explores how symmetry constraints from isometry group actions enhance fractional Sobolev embeddings on closed Riemannian manifolds, revealing improved inequalities and compactness properties.

Contribution

It establishes that G-invariant fractional Sobolev spaces embed into higher L^p spaces with improved constants, depending on orbit dimensions, advancing understanding of symmetry effects.

Findings

G-invariant fractional Sobolev spaces embed into higher L^p spaces

Compactness results depend on minimal orbit dimension

Optimal constants are characterized for improved inequalities

Abstract

In this paper, we study symmetry-improved fractional Sobolev embeddings on closed Riemannian manifolds under the action of compact isometry groups. We prove that \(G\)-invariant fractional Sobolev spaces embed into higher \(L^p\) spaces, with corresponding compactness results depending on the minimal orbit dimension. We also investigate the associated optimal constants in the improved critical inequality and in the standard critical inequality under finite-orbit symmetry.