Hi!

Let $(M,g)$ be a compact Riemannian manifold without boundary of dimension $2m$. Let

$$T:W^{2,2}(M)\rightarrow L^{2}(T^{*}M\otimes TM)$$
be a second order, linear, differential operator (coefficients in local coordinates are bounded smooth functions).

Now consider the fourth order linear operator $$\Lambda:= T^{*}T: W^{4,2}(M)\rightarrow L^{2}(M)$$

Suppose we know that: $\Lambda$ is Fredholm (finite dimensional kernel and cokernel), is elliptic with index 0 and its principal symbol is equal to that of $\Delta_g^2$. Let $$\ker(\Lambda):=span\left[\psi_1,\ldots, \psi_N \right]$$ with $\psi_i$ a $L^{2}$-orthonormal basis of $\ker(\Lambda)$. Now let $$\pi_{D}:W^{2,2}(M)\rightarrow W^{2,2}(M)$$ be the continuous projection $$\pi_{D}(f)=f-\sum_{i=1}^{N}(f,\psi_i)_{L^{2}(M)}\psi_i$$ and define $$D=\pi_{D}(W^{2,2}(M))$$

Under the above assumptions does exist $C>0$ s.t. $\forall f\in D$ we have

$$\left\| \left|T f\right|_{g} \right\|_{L^{2}(M)}\geq C\left\|f\right\|_{W^{2,2}(M)}$$

In other words, is $\left\| \left|T f\right|_{g} \right\|_{L^{2}(M)}^2$

a coercive functional on $D$?