# Application of conformal normal coordinates for higher order elliptic operator

Let $n>2$ be even. Consider a compact Riemannian manifold $(M^n,g)$ and denote with $P_g$ the critical GJMS operator.

Recall that $P_g$ is conformally invariant, i.e. $$P_{\tilde g}=e^{-nu}P_g$$ if $\tilde g=e^{2u}g$ for some $u$.

Let $H$ be given by the following expression $$H(x,y)=c_n \log(\frac{1}{r})f(r)$$ where $r=d_g(x,y)$ is the geodesic distance from $x$ to $y$ and $f(r)$ is a positive decreasing function, $f(r)=1$ in a neighborhood of $r=0$ and $f(r)=0$ for $r\geq r_{inj}$ the injectivity radius.

From Lee-Parker "The Yamabe problem" (Theorem $5.1$) there exists a metric $\tilde g$ conformal to $g$ such that $$|\tilde g(x,y)|=1+O(r^m)$$ for some $m$ big enough. In coordinates we have the following expression for the Laplace-Beltrami operator $$\Delta_{\tilde g, y}v=\frac{1}{\sqrt{|\tilde g|}}\partial_i(\tilde g^{ij}\sqrt{|\tilde g|}\partial_j v).$$ In normal conformal coordinates one has $$\tilde g^{ij}=\delta_{ij}+O(r^2)$$ $$\partial_i \tilde g^{ij}=O(r).$$ Question: prove that working in this coordinate system one has $$|P_{\tilde g} H(x,y)|\leq C r^{2-n}$$ for $r\leq Cr_{inj}$.

This is a step of the proof of Lemma 2.1 in Ndiaye "Constant Q-curvature metrics in arbitrary dimension". It is not clear to me how to do the computation to prove the estimate and why one needs conformal normal coordinates instead of geodesic coordinates.

A proof for $n=4$ is well appreciated too. In this case $$P_g v=(-\Delta_g)^2 v+\textrm{div}_g(\frac{2}{3} R_g g-2Ric_g)dv$$ where $R_g$ denotes the scalar curvature and $Ric_g$ the Ricci curvature. For $n>4$ one has $$P_g v=(-\Delta_g)^{n/2} v+l.o.t.$$