A Sharp Higher Order Sobolev Inequality on Riemannian Manifolds

Abstract

Let $m,n$ be integers such that $\frac{n}{2}>m\ge 1$ and let $(M,g)$ be a closed $n-$dimensional Riemannian manifold. We prove there exists some $B\in \mathbb{R}$ depending only on $(M,g)$, $m$, and $n$ such that for all $u\in H_m^2(M)$, $$\Vert u{\Vert }_{2^{\#}}^2\le K(m,n){\int }_M({\Delta }^{\frac{m}{2}}u{)}^{2}d{v}_g+B\Vert u{\Vert }_{H_{m-1}^2(M)}^2$$ where $2^{\#}=\frac{2n}{n-2m}$, $K(m,n)$ is the square of the best constant for the embedding $W^{m,2}({\mathbb{R}}^n)\subset L^{2^{\#}}({\mathbb{R}}^n)$, $H_m^2(M)$ is the Sobolev space consisting of functions on $M$ with $m$ weak derivatives in $L^2(M)$, and ${\Delta }^{\frac{m}{2}}=\nabla {\Delta }^{\frac{m-1}{2}}$ if $m$ is odd. This inequality is sharp in the sense that $K(m,n)$ cannot be lowered to any smaller constant. This extends the work of Hebey-Vaugon and Hebey which correspond respectively to the cases $m=1$ and $m=2$.

28 pages, comments welcome