Sharp Fractional Sobolev Embeddings on Closed Manifolds

TL;DR

This paper develops a heat-kernel based fractional Sobolev framework on closed manifolds, analyzes the critical embeddings, and establishes sharp inequalities with optimal constants and orthogonality constraints.

Contribution

It introduces an intrinsic fractional Sobolev framework on closed manifolds and derives sharp inequalities with optimal constants and orthogonality conditions.

Findings

Determined the optimal coefficient of the lower-order $L^{p}$ term.

Proved that the fully sharp $p$-power inequality cannot hold globally in the superquadratic range.

Established an almost sharp inequality with a leading constant close to the Euclidean best constant.

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.