Optimal Sobolev inequalities of high order with $L^2$-remainder

TL;DR

This paper studies the conditions under which optimal higher-order Sobolev inequalities with an $L^2$-remainder term hold on closed Riemannian manifolds, revealing geometric dependencies and sharp results for specific cases.

Contribution

It establishes precise geometric conditions for the validity of these inequalities, especially for the case $k=2$ and in low dimensions, where the inequalities are sharp.

Findings

Optimal inequalities do not hold universally for $k>1$.

Validity depends on the manifold's geometry.

Results are sharp for $k=2$ and small dimensions.

Abstract

We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal inequality does not hold in general for $k>1$. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when $k=2$ and in small dimensions.