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

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.