Lemma 5.42.2490E5EB01AE4

Proof. Let $z$ be local coordinates on $X$ and $\xi \in \mathbb C^ n$ the fiber variable with respect to a smooth orthonormal frame of the tangent bundle. Write $\rho (|\xi |^2)$ for the fixed radial averaging kernel, normalized with respect to Lebesgue measure $\mathrm{d}\xi $ and supported in $|\xi |\leq 1$. In these coordinates the averaging map $F$ gives

The map $F$ is smooth in these variables, and its fiber differential at zero is an isomorphism. Let $R_ t^*$ be the formal adjoint of $R_ t$ with respect to $\mathrm{d}V$, so that $\int _ X(R_ ta)b\, \mathrm{d}V=\int _ Xa\, R_ t^*b\, \mathrm{d}V$. We first claim that

uniformly. To see this, insert a smooth partition of unity in the variable $z$, each term being compactly supported in a chart on which the frame is defined. For fixed $\xi $, change variables $x=F(z,t\xi )$ on a neighborhood of the support of a term. The inverse map and its Jacobian have uniform Taylor expansions in $t$. The term of order zero in the transformed density is the chosen partition function at $x$, and the term of order one is linear in $\xi $, so it integrates to zero against the even radial kernel. The remaining density is bounded by $Ct^2\mathrm{d}V(x)$, since the second derivatives of all maps, partition functions and volume coefficients are bounded on these finitely many compact charts. Summing over the partition functions proves the claim.

Consequently,

By lemma 5.42.1, $R_ tu-u\geq -Kt^2$. A real function $a\geq -Kt^2$ satisfies $|a|\leq a+2Kt^2$; applying this to $a=R_ tu-u$ and integrating gives the first estimate.

For the derivative bound, change variables in the fiber to write $R_ tu(z)=\int _ Xu(x)k_ t(z,x)\, \mathrm{d}V(x)$. In local coordinates the smooth kernel $k_ t$ is supported where $|x-z|\leq Ct$ and satisfies

Indeed, the inverse fiber map and its Jacobian have uniformly bounded derivatives, and differentiating the rescaled radial argument introduces at most one factor $t^{-1}$ in addition to $t^{-2n}$. Since the kernel is smooth with compact support, there are no boundary terms. Integration over a ball of volume at most $Ct^{2n}$ gives $\int |\nabla _ zk_ t|\, \mathrm{d}V\leq C/t$, and multiplication by the uniform bound for $u$ proves the assertion. $\square$