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 $D$? 



Question about coercivity of a functional
I could not quite understand your question due to TeX problems and here is my best guess. There exists $C>0$ such that for any $f\in W^{4,2}$ orthogonal to $\ker \Lambda$ we have


$$C \Vert f\Vert\_{L^2} \leq \Vert \Lambda f\Vert\_{L^2}= \Vert Tf\Vert\_{L^2}^2.\tag{1}\label{1} $$


(For a proof of (\ref{1}) check Lemma 10.4.9 of [these notes](http://www.nd.edu/~lnicolae/Lectures.pdf).) Using elliptic estimates we deduce


$$ \Vert f\Vert\_{W^{4,2}} \leq C'(\Vert f\Vert\_{L^2}+ \Vert\Lambda f\Vert\_{L^2} ). $$


Using (\ref{1}) we deduce


$$ \Vert f\Vert\_{W^{4,2}}\leq C\_1 \Vert Tf\Vert\_{L^2}^2$$


Now let


$$\mu =\inf\bigl\lbrace \Vert Tf\Vert\_{L^2};\;\; \Vert f\Vert\_{4,2}=1,\;\; f\perp \ker \Lambda\;\bigr\rbrace\geq \frac{1}{\sqrt{C\_1}}. $$


We deduce that


$$ \mu \Vert f\Vert\_{W^{2,2}}\leq \mu \Vert f\Vert\_{W^{4,2} } \leq \Vert Tf\Vert\_{L^2},\;\;\forall f \in W^{4,2},\;\;f\perp \ker \Lambda. $$


Since $W^{4,2}$ is dense in $W^{2,2}$ we obtain the coercivity you were seeking.


