A sharp higher order Sobolev inequality on Riemannian manifolds

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 $B\in R$ depending only on $(M,g)$, m, and n such that for all $u\in H_m^2\left(M\right)$,${\Vert u\Vert }_{2^{\#}}^2\le K(m,n)\underset{M}{\int }{\left({\Delta }^{\frac{m}{2}}u\right)}^{2}d{v}_g+B{\Vert u\Vert }_{H_{m-1}^2\left(M\right)}^2$ where $2^{\#}=\frac{2n}{n-2m}$, $K(m,n)$ is the square of the best constant for the embedding $W^{m,2}\left(R^n\right)\subset L^{2^{\#}}\left(R^n\right)$, $H_m^2\left(M\right)$ is the Sobolev space consisting of functions on M with m weak derivatives in $L^2\left(M\right)$, 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 [11] and Hebey [10] which correspond respectively to the cases $m=1$ and $m=2$.