Higher order Sobolev inequality

Question

Let $(M,g)$ be a closed, Riemannian manifold of dimension $n>4$. Let $K$ be the best constant for the Sobolev inequality

$||u||^2_p \leq K \int_{{\Bbb{R}}^n} (\Delta u)^2 dx,$

where $p=\frac{2n}{n-4}$. Here we are assuming that $u \in H^2(\Bbb{R}^n)$, so constants are excluded. Then by work in Djadli et al. we have it that there exists a constant $B$ such that for any $\epsilon$ greater than zero, we have it that

$||u||_p^2 \leq (K+\epsilon) \int_M (\Delta u)^2 + B(|\nabla u|^2 + u^2) dv_g,$

for all $u \in H^2(M)$. My question is whether or not there is a $H^3(M)$ generalization of this result -- something like the following: for every $\epsilon > 0$ there exists a constant $B$ such that

$||u||_q^2 \leq (M + \epsilon) \int_M |\nabla \Delta u|^2 + B((\Delta u)^2 + |\nabla u|^2 + u^2) dv_g,$

for all $u \in H^3(M)$, where $q = \frac{2n}{n-6}$, $n>6$, and $M$ is the best constant for the Sobolev inequality on $\Bbb{R}^n$,

$||u||^2_q \leq M \int_{{\Bbb{R}}^n} |\nabla \Delta u|^2 dx.$

I tried to modify the proof given by Djadli et al., but it breaks down because $\nabla \Delta u$ will involve second order derivatives of the metric tensor.

Answer 1

The desired inequality is correct. It is easily generalized from Aubin's proof (which is given in Lee&Parker's "Yamabe Problem") of the classical Sobolev inequality for Riemannian manifolds. One simply use a partition of unity argument to transfer the $\Bbb{R}^n$ result to the compact manifold. This requires numerous applications of the Cauchy-Schwarz inequality and the Cauchy inequality after one expands the derivatives of the product of the function with the partition functions. The key idea is that you can estimate all of the second, first, and zero order terms in a crude fashion, because the third order term is the only one that needs to be estimated carefully. This process seems like it could be generalized easily to get similiar Sobolev inequalities for $H^k(M)$ where $k>3$.