A Sharp Higher Order Sobolev Inequality on Riemannian Manifolds

Let $m, n$ be integers such that $\frac{n}{2} > m \geq 1$ 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 $u \in H_m^2(M)$, $$\lVert u \rVert_{2^\#}^2 \leq K(m,n) \int_M (\Delta^\frac{m}{2} u)^2 dv_g + B \lVert u \rVert_{H_{m-1}^2(M)}^2$$ 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 $m=2$.