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

Hao Tan, Zhipeng Yang

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.

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

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 -invariant fractional Sobolev spaces embed into higher 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.

Paper Structure

This paper contains 5 sections, 20 theorems, 208 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary results
  3. Proof of Theorem \ref{['thm1.1']}
  4. Proof of Theorem \ref{['thm1.2']}
  5. Proof of Theorem \ref{['thm1.4']}

Key Result

Theorem 1.1

Let $(M,g)$ be a closed Riemannian $n$-manifold, let $s\in(0,1)$, let $q\in[1,\infty)$, and let $G\subset \operatorname{Isom}_g(M)$ be compact. Assume that and set Then $k\ge 1$, and the following assertions hold.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 2.1: Tan2025SharpFS
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 30 more