Sharp Fractional Sobolev Embeddings on Closed Manifolds

TL;DR

This work develops an intrinsic, heat-kernel driven framework for fractional Sobolev spaces on closed Riemannian manifolds and analyzes sharp Sobolev embeddings in the fractional setting. By defining $W^{s,p}(M)$ via heat-based kernels and establishing intrinsic constants, it separates Euclidean-like leading behavior from curvature-driven lower-order terms. The authors prove exact sharpness results for the linear $L^p$ term and almost-sharp leading constants for the $L^{p_s^*}$ embedding, including improvements under finitely many orthogonality constraints, with a concentration-compactness approach that bypasses full Euclidean extremal classifications. These results extend the classical local manifold Sobolev theory to nonlocal settings, with implications for nonlocal equations and geometric analysis on manifolds.

Abstract

We develop an intrinsic, heat-kernel based fractional Sobolev framework on closed Riemannian manifolds and study the critical fractional Sobolev embedding. We determine the optimal coefficient of the lower-order $L^{p}$ term and prove that the fully sharp $p$-power inequality cannot hold globally in the superquadratic range. We further establish an almost sharp inequality whose leading constant is arbitrarily close to the Euclidean best constant, and we derive improved inequalities under finitely many orthogonality constraints with respect to sign-changing test families.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• Proposition 2.2
• Definition 3.1: Spectral fractional Laplacian
• Remark 3.2
• Definition 3.3
• Theorem 3.4
• proof