16.17 Smoothing and its joint complex HessianB73B45463F24

Proof. We construct the smoothing in Step 1. In Steps 2–4 we compute, for a smooth plurisubharmonic local potential $p$, the joint complex Hessian of $\Psi _ p$ up to a remainder of high order in the radius; the result is the identity (16.17.16). From this identity we derive the joint estimates, first for smooth potentials (Step 5) and then in general (Step 6), and Step 7 proves the remaining assertions. All bounds are uniform over the centers, because the constructions are made in the charts of a fixed finite atlas, with coefficients bounded uniformly in the center.

Step 1: construction of the smoothing. We construct a smooth map $F(x,\zeta )\in X$, defined for small tangent vectors $\zeta \in T_ xX$, from the holomorphic parts of the jets of geodesics, first in coordinates and then globally; $\Psi _ u$ is then an average of $u\circ F$.

We begin with uniform normal coordinates. Fix the jet order $N=8$. For $x\in X$ we use the normal coordinates for $\omega $ at $x$ of lemma 16.15.3, applied in a chart of a fixed finite atlas that contains $x$ in a fixed compact subset; in them $h(0)=I$ and $\mathrm dh(0)=0$. The coordinate change there is a polynomial map of degree at most two. Its linear part $L_ x$ satisfies $L_ x^{\mathrm T}h(x)\overline{L_ x}=I$, where $h(x)$ is the coefficient matrix in the chart of the atlas, so $L_ x$ and $L_ x^{-1}$ are bounded uniformly in $x$. The condition $\mathrm dh(0)=0$ is a linear equation for the quadratic part, whose coefficient matrix $h(x)\overline{L_ x}$ is invertible and whose right side is built from $L_ x$ and the first derivatives of $h$ at $x$; so the quadratic part is bounded uniformly in $x$ as well. Hence these coordinates are defined on balls of a fixed size, and all finite jets of $h$ in them are bounded uniformly in $x$.

Next we define the jet map, in arbitrary holomorphic coordinates. Let $\Gamma ^ m_{jl}$ be given by $\sum _ m\Gamma ^ m_{jl}h_{m\bar k}=\partial _ jh_{l\bar k}$. Thus $\Gamma (z)$ is a bilinear map $\mathbb C^ n\times \mathbb C^ n\to \mathbb C^ n$, symmetric because $\widehat\omega $ is closed, depending smoothly on $z$, and $\Gamma (0)=0$ in normal coordinates. Consider the geodesic equation

for formal power series $\gamma (t)=z+\sum _{k\geq 1}c_ kt^ k/k!$ in a real variable $t$, with $c_ k\in \mathbb C^ n$; here $\Gamma (\gamma (t))$ is the composition of the Taylor series of $\Gamma $ at $z$ with the series $\gamma (t)-z$, which has no constant term. Prescribe $c_1=\zeta $. Comparing the coefficients of $t^ k$ in the equation determines $c_{k+2}$ as a polynomial in $c_1,\ldots ,c_{k+1}$ whose coefficients are derivatives of $\Gamma $ at $z$ of order at most $k$. By induction, $c_ k$ is a polynomial in $\zeta $ and $\bar\zeta $ with coefficients depending smoothly on $z$, and it is homogeneous of degree $k$: for real $\lambda $ the series $\gamma (\lambda t)$ is the formal solution with initial velocity $\lambda \zeta $, so that $c_ k(\lambda \zeta )=\lambda ^ kc_ k(\zeta )$. Let $c_ k^{\mathrm h}(z,\zeta )$ be the sum of the monomials of $c_ k$ that contain no $\bar\zeta $, and define

a holomorphic polynomial in $\zeta $ with coefficients depending smoothly on $z$. Since $c_2=-\Gamma (z)(\zeta ,\zeta )$, in normal coordinates at $x$ we have $J_0(\zeta )=\zeta +O(|\zeta |^3)$.

These jets do not depend on the coordinates. Let $y=\phi (z)$ be a holomorphic change of coordinates, with derivative $\Phi =D\phi $, and let $\widetilde h$, $\widetilde\Gamma $ and $\widetilde J$ be the corresponding objects in the coordinates $y$. Then $h_{j\bar k}=\sum _{a,b}\Phi ^ a_ j\, \widetilde h_{a\bar b}(\phi )\, \overline{\Phi ^ b_ k}$, and since $\Phi $ is holomorphic,

Comparing with the definition of $\Gamma $, and using that the matrix $(\sum _ b\widetilde h_{a\bar b}\overline{\Phi ^ b_ k})_{a,k}$ is invertible, we obtain $\Phi \, \Gamma (\zeta ,\zeta ')=D^2\phi (\zeta ,\zeta ')+\widetilde\Gamma (\phi )(\Phi \zeta ,\Phi \zeta ')$ for all $\zeta ,\zeta '\in \mathbb C^ n$. Let $\gamma $ be a formal solution in the coordinates $z$. The second derivative of $\phi \circ \gamma $ is $\Phi (\gamma )\ddot\gamma +D^2\phi (\gamma )(\dot\gamma ,\dot\gamma )$, because $\Phi $ is holomorphic, and therefore

By uniqueness, $\phi \circ \gamma $ is the formal solution with initial point $\phi (z)$ and initial velocity $\Phi (z)\zeta $. By homogeneity we may put $t=1$ degree by degree, and we obtain an identity between formal power series in $\zeta $ and $\bar\zeta $. Call a monomial in $\zeta $ and $\bar\zeta $ discardable if it contains a factor $\bar\zeta _ m$ or has degree at least $N+1$; the product of a discardable monomial with any monomial is discardable. The two sides of this identity, the formal solution in the coordinates $y$ and the series $\phi \circ \gamma $, differ from $\widetilde J_{\phi (z)}(\Phi (z)\zeta )$ and from $\phi (J_ z(\zeta ))$, respectively, by series of discardable monomials; for $\phi \circ \gamma $ we use the Taylor expansion of the holomorphic map $\phi $ at $J_ z(\zeta )$, each of whose terms beyond the first contains the discardable difference as a factor. The two holomorphic functions of $\zeta $ just named therefore have the same Taylor coefficients of degree at most $N$ at $\zeta =0$, and Taylor's formula gives

where the remainder is a smooth function vanishing to order $N+1$ at $\zeta =0$, with bounds for its derivatives that are locally uniform in $z$.

We now glue the local jet maps. Choose finitely many holomorphic charts $\kappa _\nu \colon \Omega _\nu \to \mathbb C^ n$, $1\leq \nu \leq \nu _0$, and open sets $V_\nu \Subset \Omega _\nu $ covering $X$. For $x\in \Omega _\nu $ and small $\zeta \in T_ xX$ put $F_\nu (x,\zeta )=\kappa _\nu ^{-1}\bigl (J^\nu _{\kappa _\nu (x)}(D\kappa _\nu (x)\zeta )\bigr )$, where $J^\nu $ is the jet map in the coordinates $\kappa _\nu $. By (16.17.7), in every holomorphic chart, $F_\nu $ and $F_\mu $ differ by $O(|\zeta |^{N+1})$ where both are defined. We say that a smooth map $\widehat F(x,\zeta )$, defined for $x$ in an open set $O$ and $|\zeta |$ below a positive constant, has the geodesic jets of order $N$ if in every chart $\kappa _\nu $ we have $\widehat F-F_\nu =O(|\zeta |^{N+1})$, with bounds that are uniform on compact subsets of $O\cap \Omega _\nu $.

We construct inductively such maps $\widehat F_ k$ on open sets $O_ k\supset \overline{V_1\cup \cdots \cup V_ k}$. Let $\widehat F_1=F_1$ and $O_1=\Omega _1$. Given $\widehat F_{k-1}$, choose an open set $O'$ with $\overline{V_1\cup \cdots \cup V_{k-1}}\subset O'\Subset O_{k-1}$, a function $\vartheta \in C^\infty _ c(\Omega _ k)$ with $0\leq \vartheta \leq 1$ that equals one on a relatively compact open neighborhood $O''\subset \Omega _ k$ of the compact set $\overline{V_ k}\setminus O'$ (lemma 2.5.1(4)), and an open set $\Omega '$ with $\operatorname{Supp}\vartheta \subset \Omega '\Subset \Omega _ k$. Put $O_ k=O'\cup O''$, and define $\widehat F_ k=\widehat F_{k-1}$ on $O'\setminus \operatorname{Supp}\vartheta $, $\widehat F_ k=F_ k$ on $O''$, and

For $x$ in the compact set $\overline{O'}\cap \overline{\Omega '}$ and small $\zeta $, the points $\widehat F_{k-1}(x,\zeta )$ and $F_ k(x,\zeta )$ lie in a small ball about $\kappa _ k(x)$ in the chart $\kappa _ k$, so the last formula is defined. The three sets cover $O_ k$, and the three definitions agree on the overlaps, because $\vartheta =0$ outside $\operatorname{Supp}\vartheta $ and $\vartheta =1$ on $O''$. In the chart $\kappa _ k$ the last formula gives $\widehat F_ k-F_ k=(1-\vartheta )(\widehat F_{k-1}-F_ k)=O(|\zeta |^{N+1})$, so $\widehat F_ k$ has the geodesic jets of order $N$.

The resulting map $F=\widehat F_{\nu _0}$ is defined on $X$ for $|\zeta |<r_1$, with a fixed $r_1>0$. Since $F(x,0)=x$ and $D_\zeta F(x,0)$ is the identity, we may decrease $r_1$ so that, in each chart, $\| D_\zeta F-I\| \leq \frac12$ for $|\zeta |<r_1$; then each fiber map $\zeta \mapsto F(x,\zeta )$ is injective, because $|F(x,\zeta )-F(x,\zeta ')-(\zeta -\zeta ')|\leq \frac12|\zeta -\zeta '|$, and by the inverse function theorem it is a diffeomorphism onto its image. In the normal coordinates at a center $x$, (16.17.7) shows that $F(x',\zeta )$ differs from the jet map $J_{z'}(\zeta )$ of these coordinates by $O(|\zeta |^{N+1})$, where $z'$ is the coordinate of $x'$. Since $J_0(\zeta )=\zeta +O(|\zeta |^3)$, we obtain at the center

Now we define the smoothing. Let $\chi $ be the weight of the statement, and let $\chi _1(t)=-C_ n\mathrm e^{-1/(1-t)}$ for $t<1$ and $\chi _1(t)=0$ otherwise. Both functions are smooth on $\mathbb R$, $\chi _1'=\chi $, and $-\chi _1/\chi =(1-t)^2$ for $t<1$. Define

where $\mathrm d\zeta _ h$ is the Lebesgue measure of the Hermitian space $(T_ xX,h)$; in coordinates it is $\det h$ times the Lebesgue measure of the components of $\zeta $, and in unitary coordinates of $T_ xX$ the kernel has total mass $\int \chi (|\zeta |^2)\, \mathrm d\zeta =1$. The substitution $\zeta \mapsto \mathrm e^{\mathrm it}\zeta $ shows that $\Psi _ u$ is invariant under rotations of $w$. For $w\neq 0$, the change of variables $y=F(x,w\zeta )$ in the fiber, whose inverse depends smoothly on $(x,w,y)$ by the inverse function theorem, writes $\Psi _ u(x,w)$, in a chart around $x$, as the integral of $u$ against a smooth nonnegative kernel supported in a ball of radius $C|w|$ about $x$. Differentiation under the integral sign shows that $\Psi _ u$ is smooth for $w\neq 0$ even when $u$ has poles.

Step 2: a holomorphic parametrization of the fiber. Throughout Steps 2–4 we fix a center, the origin of its normal coordinates $z$, and a smooth plurisubharmonic function $p$ on a neighborhood of $\overline{B(0,3r)}$, where $r=|w|$ and $0<|w|<r_0$; Step 6 removes the smoothness. In this step we rewrite $\Psi _ p$ so that $p$ is composed with a map that is holomorphic in all variables. We write $F(z,\xi )$ for $F(x',\zeta )$ when $x'$ has coordinate $z$ and $\zeta =\sum _ m\xi _ m\partial /\partial z_ m$. We use multi-indices $I,J$ with $2\leq |I|,|J|\leq N$ and write $1_ l$ for the $l$th coordinate multi-index; a product such as $I_ ls^{I-1_ l}$ is zero when $I_ l=0$, and the same convention applies to conjugates and to $J$.

We first separate the holomorphic part of $F$. Let $a(z,\rho )$ be the polynomial formed by the holomorphic monomials of the Taylor expansion of $F$ at $(0,0)$ that have degree at most two in $z$ and at most $N$ in $\xi $, with $\xi $ renamed $\rho $. Since $F(z,0)=z$ and $D_\xi F(z,0)$ is the identity, $a(z,0)=z$ and $\partial _\rho a(z,0)=I$. Hence, for small $z$, the holomorphic map $a(z,\cdot )$ has a holomorphic inverse near $0$, and we define

We extend the notion of a discardable monomial to the variables $(z,\xi )$: a monomial in $z,\bar z,\xi ,\bar\xi $ is discardable if it contains a factor $\bar z_ j$ or $\bar\xi _ m$, has degree at least three in $z$, or has degree at least $N+1$ in $(\xi ,\bar\xi )$. Again the product of a discardable monomial with any monomial is discardable. By construction, the Taylor expansion of $F(z,\xi )-a(z,\xi )$ at $(0,0)$ consists of discardable monomials. Since $\rho (z,\xi )-\xi $ is the integral over $t\in [0,1]$ of the derivative of $a(z,\cdot )^{-1}$ at $a(z,\xi )+t(F(z,\xi )-a(z,\xi ))$, applied to $F(z,\xi )-a(z,\xi )$, the same holds for $\rho (z,\xi )-\xi $. Moreover, for each fixed $z$, the Taylor polynomial of degree $N$ of $F(z,\cdot )$ at $0$ is holomorphic, because it is that of $J_ z$; since $a(z,\cdot )^{-1}$ is holomorphic, the same holds for $\rho (z,\cdot )$. Together with $\rho (z,0)=0$ and $\partial _\xi \rho (z,0)=I$, Taylor's formula in $\xi $ therefore gives

with smooth vector-valued coefficients $\rho _ I$ and a smooth remainder $R$ whose derivatives of order at most seven, with $k$ of them in $\xi $ or $\bar\xi $, are bounded by $C|\xi |^{N+1-k}$. The monomials $\xi ^ I$ and $z_ j\xi ^ I$ are not discardable, so $\rho _ I(0)=0$ and $\partial _{z_ j}\rho _ I(0)=0$. We put

vectors with components $d_{Ikm}$ and $e_{Ijkm}$.

Put $\tilde\rho (z,w,\xi )=w^{-1}\rho (z,w\xi )$, so that

By the chain rule and the bounds for $R$, for $|\xi |\leq 3$ every derivative of $\tilde R$ of order at most two in $(z,\bar z)$, at most three in $(\xi ,\bar\xi )$ and $k\leq 2$ in $(w,\bar w)$ is bounded by $Cr^{N-k}$. Since also $\rho _ I(z)=O(|z|)$, we have $\| \partial _\xi \tilde\rho -I\| +\| \partial _{\bar\xi }\tilde\rho \| \leq \frac14$ for $|\xi |\leq 3$ when $|z|$ and $r$ are small. As above, $\xi \mapsto \tilde\rho (z,w,\xi )$ is then a diffeomorphism of $\{ |\xi |<3\} $ onto an open set, which contains $\{ |s|\leq 2\} $ by the contraction mapping principle, applied to $\xi \mapsto s+\xi -\tilde\rho (z,w,\xi )$ on $\{ |\xi |\leq 8/3\} $ (note that $\tilde\rho (z,w,0)=0$); we denote its inverse by $s\mapsto \xi (z,w,s)$. Since $a(z,w\tilde\rho (z,w,\xi ))=F(z,w\xi )$, the substitution $s=\tilde\rho (z,w,\xi )$ gives

where $\mathcal Q=|\xi (z,w,s)|^2_{h(z)}$, $\mathcal J$ is $\det h(z)$ times the real Jacobian determinant of $s\mapsto \xi (z,w,s)$, and $W$ is extended by zero outside the image. On the support of $\chi (|\xi |_ h^2)$ we have $|\xi |<2$ and hence $|s|<2$ for small $|z|$ and $r$, so $W$ is smooth and vanishes for $|s|\geq 2$. Thus the argument $a(z,ws)$ of $u$ is holomorphic in $(z,w,s)$, and the weight $W$ carries the nonholomorphic dependence on the base point.

Step 3: the weight at the center. We compute the first and the mixed second derivatives of $W$ in $z$ at $z=0$, up to errors of order $r^ N$, and write them as divergences in $s$; the result is (16.17.11). For functions of $(w,s)$ with $0<|w|<r_0$ and $|s|\leq 2$, we write $O_ N$ for any function whose derivatives of order at most two in $(s,\bar s)$ and $k\leq 2$ in $(w,\bar w)$ are bounded by $Cr^{N-k}$, where $r=|w|$. Put

vectors that are holomorphic in $s$; for a holomorphic vector field $b(s)$ write $\operatorname {div}b=\sum _ m\partial _{s_ m}b_ m$, $b\cdot \bar s=\sum _ mb_ m\bar s_ m$, and $b\cdot \bar b'=\sum _ mb_ m\overline{b'_ m}$.

We begin with the derivatives of $\xi $. We differentiate the identity $\tilde\rho (z,w,\xi (z,w,s))=s$ in $\bar z_ k$, in $z_ j$, and then in $z_ j$ and $\bar z_ k$, and evaluate at $z=0$, where $\rho _ I=0$ and $\partial _{z_ j}\rho _ I=0$. In each case the derivative of $\xi $ to be determined appears multiplied by $\partial _\xi \tilde\rho =I+\partial _\xi \tilde R$, and the remaining terms are the explicit terms $\partial _{\bar z_ k}\rho _ Iw^{|I|-1}\xi ^ I$ and $\partial _{z_ j}\partial _{\bar z_ k}\rho _ Iw^{|I|-1}\xi ^ I$, terms containing a derivative of $\tilde R$, and terms containing the factor $\partial _{z_ j}\xi $. The conjugates of these derivatives of $\xi $ also occur, multiplied by $\partial _{\bar\xi }\tilde R$; since $\partial _\xi \tilde R$ and $\partial _{\bar\xi }\tilde R$ are bounded by $Cr^ N$, each of these linear systems is a small perturbation of the identity, and its solutions are bounded uniformly. Using this, the bounds for $\tilde R$, and the fact that $\beta ^ k$ and $\gamma ^{jk}$ have bounded derivatives, we find successively that at $z=0$

the bounds for one further derivative in $s$ follow in the same way, because $\tilde R$ has bounded derivatives of order three in $\xi $. Differentiating the identity in $\bar s$ instead, we find $\partial _{\bar s}\xi =-(\partial _\xi \tilde\rho )^{-1}\partial _{\bar\xi }\tilde R\; \overline{\partial _ s\xi }$, because the coefficients of (16.17.9) do not involve $\bar\xi $; hence $\partial _{\bar s}\xi $ and its derivatives $\partial _{z_ j}$, $\partial _{\bar z_ k}$ and $\partial _{z_ j}\partial _{\bar z_ k}$ are $O_ N$ at $z=0$.

Next, $\mathcal Q=\sum _{l,m}h_{l\bar m}(z)\xi _ l\bar\xi _ m$, where $h(0)=I$ and $\mathrm dh(0)=0$; we put $c_{jklm}=-\partial _{z_ j}\partial _{\bar z_ k}h_{l\bar m}(0)$. By the product rule, (16.17.10) and $\partial _{z_ j}\bar\xi =\overline{\partial _{\bar z_ j}\xi }$, at $z=0$

and $\partial _{z_ j}\mathcal Q=\overline{\partial _{\bar z_ j}\mathcal Q}$, since $\mathcal Q$ is real.

For the Jacobian factor $\mathcal J$, let $M=\partial _ s\xi $. The real Jacobian determinant of $s\mapsto \xi $ is the determinant of the matrix $\bigl (\begin{smallmatrix}M& \partial _{\bar s}\xi \\ \overline{\partial _{\bar s}\xi }& \overline M\end{smallmatrix}\bigr )$, that is, $|\det M|^2$ plus a polynomial in the entries of these blocks each of whose terms contains an entry of $\partial _{\bar s}\xi $ or of its conjugate; as shown above, these entries and their relevant derivatives are $O_ N$ at $z=0$. Differentiating (16.17.10) in $s$, we have at $z=0$: $M=I+O_ N$, $\partial _{z_ j}M=O_ N$, $\partial _{\bar z_ k}M=-\partial _ s\beta ^ k+O_ N$ and $\partial _{z_ j}\partial _{\bar z_ k}M=-\partial _ s\gamma ^{jk}+O_ N$. Jacobi's formula $\partial \det M=\det M\operatorname {tr}(M^{-1}\partial M)$, differentiated once more, gives $\partial _{\bar z_ k}\det M=-\operatorname {div}\beta ^ k+O_ N$, $\partial _{z_ j}\det M=O_ N$ and $\partial _{z_ j}\partial _{\bar z_ k}\det M=-\operatorname {div}\gamma ^{jk}+O_ N$. Writing $|\det M|^2=\det M\cdot \overline{\det M}$ and using $\det h(0)=1$, $\mathrm d\det h(0)=0$ and $\partial _{z_ j}\partial _{\bar z_ k}\det h(0)=-\sum _ lc_{jkll}$, we obtain at $z=0$

Finally we differentiate $W=\chi (\mathcal Q)\mathcal J$. By the chain rule, $\partial _{\bar z_ k}W=\chi '(\mathcal Q)\mathcal J\, \partial _{\bar z_ k}\mathcal Q+\chi (\mathcal Q)\partial _{\bar z_ k}\mathcal J$, and $\partial _{z_ j}\partial _{\bar z_ k}W$ is the sum of $\chi ''\mathcal J\partial _{z_ j}\mathcal Q\, \partial _{\bar z_ k}\mathcal Q$, $\chi '\bigl (\mathcal J\partial _{z_ j}\partial _{\bar z_ k}\mathcal Q+\partial _{z_ j}\mathcal Q\, \partial _{\bar z_ k}\mathcal J+\partial _{\bar z_ k}\mathcal Q\, \partial _{z_ j}\mathcal J\bigr )$ and $\chi \, \partial _{z_ j}\partial _{\bar z_ k}\mathcal J$, with $\chi $ and its derivatives evaluated at $\mathcal Q$. Since $\mathcal Q=|s|^2+O_ N$ at $z=0$, we may evaluate them at $|s|^2$ instead, at the cost of an error $O_ N$. From now on $\chi ,\chi ',\chi ''$ denote these functions evaluated at $|s|^2$, and we recall that $\partial _{s_ m}|s|^2=\bar s_ m$. Inserting the formulas above, we find at $z=0$ that $\partial _{\bar z_ k}W=-\chi '\beta ^ k\cdot \bar s-\chi \operatorname {div}\beta ^ k+O_ N$. In the mixed derivatives we group the terms. The terms containing $c$ give $-\sum _{l,m}c_{jklm}(\chi 's_ l\bar s_ m+\chi \delta _{lm})$. The terms containing $e$ give $-\chi '\gamma ^{jk}\cdot \bar s-\chi \operatorname {div}\gamma ^{jk}$ and its counterpart with $\overline{\gamma ^{kj}}$. The terms quadratic in $d$ are

Since $\beta ^ k$ is holomorphic and $\overline{\beta ^ j}$ antiholomorphic in $s$, we have $\sum _ l\partial _{\bar s_ l}(\chi \beta ^ k_ m\overline{\beta ^ j_ l})=\beta ^ k_ m\bigl (\chi '\, \overline{\beta ^ j}\cdot s+\chi \overline{\operatorname {div}\beta ^ j}\bigr )$, and applying $\sum _ m\partial _{s_ m}$ to this produces exactly the five terms above. The first two groups are divergences as well: $\partial _{s_ m}(\chi s_ l)=\chi 's_ l\bar s_ m+\chi \delta _{lm}$, and, since $\gamma ^{jk}$ is holomorphic and $\overline{\gamma ^{kj}}$ antiholomorphic in $s$, $\sum _ m\partial _{s_ m}(\chi \gamma ^{jk}_ m)=\chi '\gamma ^{jk}\cdot \bar s+\chi \operatorname {div}\gamma ^{jk}$ and $\sum _ m\partial _{\bar s_ m}(\chi \overline{\gamma ^{kj}_ m})=\chi '\, \overline{\gamma ^{kj}}\cdot s+\chi \overline{\operatorname {div}\gamma ^{kj}}$. We have proved that at $z=0$

and $\partial _{z_ j}W=\overline{\partial _{\bar z_ j}W}$, since $W$ is real. Here $W$ and the explicit terms vanish for $|s|\geq 2$, so the remainders do as well.

Step 4: an identity for the joint Levi form. We compute $\mathcal D\Psi _ p[v,\eta ]^2$ at $z=0$ by differentiating under the integral sign, inserting (16.17.11) and integrating by parts in $s$; the result is the identity (16.17.16). Put $G(z,w,s)=p(a(z,ws))$ and $\mathcal L=\sum _ jv_ j\partial _{z_ j}+\eta \, \partial _ w$, so that $\mathcal Df[v,\eta ]^2=\mathcal L\bar{\mathcal L}f$ for every smooth function $f$ of $(z,w)$. At $z=0$ write $D_\rho a=\partial _\rho a(0,ws)$, let $\partial _{\rho _ m}a$ denote its $m$th column, and put

these are holomorphic in $s$. Let $P=(p_{l\bar m})\geq 0$ be the complex Hessian of $p$ at $a(0,ws)$, and write $P(y,\bar y')=\sum _{l,m}p_{l\bar m}y_ l\bar y'_ m$, so that $P[y]^2=P(y,\bar y)$. Since $a(z,ws)$ is holomorphic in $(z,w,s)$, we have $\mathcal LG=\sum _ lp_ l(a)\, (\mathcal La)_ l$, where $p_ l=\partial p/\partial z_ l$, the vector $\mathcal La$ is holomorphic and equals $Y$ at $z=0$, and $\partial _{s_ m}[a(0,ws)]=w\, \partial _{\rho _ m}a$. Hence, at $z=0$,

Next we insert the expansion of the weight. The integrand $GW$ is smooth with support in $\{ |s|\leq 2\} $, so we may differentiate under the integral sign. Since $G$ and $W$ are real, $\bar{\mathcal L}G\, \mathcal LW$ is the conjugate of $\mathcal LG\, \bar{\mathcal L}W$, and at $z=0$

Here, at $z=0$, $\bar{\mathcal L}W=\sum _ k\bar v_ k\partial _{\bar z_ k}W+\bar\eta \partial _{\bar w}W$ and

where the derivatives in $w$ may be taken after putting $z=0$. We insert (16.17.11). The explicit terms there do not depend on $w$ except through $\beta ^ k$ and $\gamma ^{jk}$, and $\partial _ w\beta ^ k=\sum _ I(|I|-1)d_{Ik}w^{|I|-2}s^ I$; each derivative of a remainder in $w$ costs one power of $r$. Thus at $z=0$

where $[\partial _{z_ j}\partial _{\bar z_ k}W]^\flat $ denotes the explicit part of (16.17.11), and $\delta _0,\delta _1,\delta _2$ vanish for $|s|\geq 2$ and satisfy

Accordingly $\mathcal D\Psi _ p[v,\eta ]^2=T_1+T_2+T_3+E$, where $T_1=\int P[Y]^2\chi \, \mathrm ds$, $T_2$ and $T_3$ are the contributions of the explicit parts of $\bar{\mathcal L}W$ and $\mathcal L\bar{\mathcal L}W$, and

We evaluate $T_2$ and $T_3$ by integrating by parts in $s$, using the following divergence formulas: for a multi-index $I$ with $|I|\geq 1$,

Indeed, $\partial _{s_ m}\chi _1(|s|^2)=\chi (|s|^2)\bar s_ m$ because $\chi _1'=\chi $, the monomial $\bar s^{I-1_ m}$ is antiholomorphic, and Euler's identity $\sum _ mI_ m\bar s_ m\bar s^{I-1_ m}=|I|\bar s^ I$ gives the first formula; the second is its complex conjugate. Similarly $\chi (|s|^2)s_ l=\partial _{\bar s_ l}\chi _1(|s|^2)$. The integrations by parts are justified as follows. If $f$ and $g$ are continuously differentiable on a neighborhood of the support of $g$, and this support is compact, then $\int f\, \partial _{s_ m}g\, \mathrm ds=-\int g\, \partial _{s_ m}f\, \mathrm ds$, and similarly for $\partial _{\bar s_ m}$. Since $\partial _{s_ m}=\frac12(\partial _{\operatorname{Re}s_ m}-\mathrm i\partial _{\operatorname{Im}s_ m})$, this follows from Fubini's theorem and the fundamental theorem of calculus, applied to the compactly supported function $fg$. In all integrations by parts below, one of the two factors is a product of $\chi (|s|^2)$ or $\chi _1(|s|^2)$ with a polynomial, or one of the remainders in (16.17.13), hence smooth with support in $\{ |s|\leq 2\} $, so no boundary terms occur.

We now compute $T_2$. By (16.17.15),

where $\sigma ^ k_{ml}$ is holomorphic in $s$. Integrating by parts in $\bar s_ l$ and using (16.17.12) and $\partial _{s_ m}\chi _1(|s|^2)=\chi \bar s_ m$, we get

We multiply by $\bar v_ k$ and sum over $k$ and $m$. To write the result, we define vectors $\Xi $ and $\Theta $ by

where the factor $(I-1_ m)_ l\bar s^{I-1_ m-1_ l}$ is zero when $(I-1_ m)_ l=0$. Then $\bar w\sum _{k,m}\bar v_ k\bar s_ m\sigma ^ k_{ml}=r^2\, \overline{\Xi ^ l}$ and $\bar w\sum _{k,m}\bar v_ k\partial _{s_ m}\sigma ^ k_{ml}=r^2\, \overline{\Theta ^ l}$, and $P$ is antilinear in its second argument. Therefore

Next we compute $T_3$. By (16.17.15), each explicit term of $\mathcal L\bar{\mathcal L}W$ is a sum of second derivatives $\partial _{s_ m}\partial _{\bar s_ l}$ of weights, and two integrations by parts move these derivatives onto $G$. For the curvature term we use $\partial _{s_ m}(\chi s_ l)=\partial _{s_ m}\partial _{\bar s_ l}\chi _1$ and (16.17.12), and we let $C^ v$ be the matrix with entries $C^ v_{ml}=\sum _{j,k}c_{jklm}v_ j\bar v_ k$:

Since $h$ is Hermitian, $\overline{c_{jklm}}=c_{kjml}$; hence $C^ v$ is Hermitian, and $\| C^ v\| \leq C|v|^2$ for its operator norm. For the terms containing $\gamma ^{jk}$ and $\partial _ w\beta ^ k$, (16.17.15) writes $\chi \gamma ^{jk}_ m=\sum _ l\partial _{\bar s_ l}(\chi _1\tau ^{jk}_{ml})$ and $\chi \partial _ w\beta ^ k_ m=\sum _ l\partial _{\bar s_ l}(\chi _1\partial _ w\sigma ^ k_{ml})$, with $\sigma ^ k_{ml}$ as above and

Two integrations by parts give the contribution

The terms containing $\overline{\gamma ^{kj}}$ and $\overline{\partial _ w\beta ^ j}$ are treated with the conjugate formula in (16.17.15), and after renaming $j\leftrightarrow k$ they give the same expression with the matrix $B$ replaced by its conjugate transpose $B^*$. Finally, with $\beta =\sum _ k\bar v_ k\beta ^ k$, two integrations by parts turn the last term of (16.17.11) into

For a matrix $M$ in the fiber variables write

so that $P[Da(yy'^*)]=P(D_\rho a\, y,\overline{D_\rho a\, y'})$ for vectors $y$ and $y'$. Since $\chi $ and $\chi _1$ vanish outside the unit ball, on which $\chi >0$, we have shown that $T_3=r^2\int P[Da(U)]\chi \, \mathrm ds$, where, on the unit ball, $U=-\frac{\chi _1}{\chi }(C^ v+B+B^*)+\beta \beta ^*$, with the quotient $\chi _1/\chi $ evaluated at $|s|^2$. Explicitly, $U$ is the Hermitian part of the matrix $U'=-\frac{\chi _1}{\chi }(C^ v+2B)+\beta \beta ^*$, whose entries are

Finally we add $T_1$, $T_2$ and $T_3$. Completing the square, $P[Y]^2+2r^2\operatorname{Re}P(Y,\overline{D_\rho a\, \Xi })=P[Y+r^2D_\rho a\, \Xi ]^2-r^4P[D_\rho a\, \Xi ]^2$, where $Y+r^2D_\rho a\, \Xi =Da(v,\eta s+r^2\Xi )$. Moreover, the real part of $P(Y,\overline{D_\rho a\, \Theta })$ equals that of $P(D_\rho a\, \Theta ,\overline Y)$, because $P$ is Hermitian. We have therefore proved the identity

Here $Da$ is the differential of $a$ at $(0,ws)$, $P[Da(\cdot )]$ is the contraction defined above, and the remainder $E$ is given by (16.17.14).

Step 5: the joint estimate for smooth potentials. In this step we assume that $p=u+q$ is smooth near $\overline{B(0,3r)}$. We derive (16.17.2) from (16.17.16) and prove that $H_ u$ is a strictly convex function of $\log r$. We choose $q$ as a fixed local potential of $\alpha $ on one of finitely many coordinate balls covering $X$ (lemma 5.4.2); by a Lebesgue number argument, the normal ball of a fixed radius about each center lies in one of them. This choice is harmless, because $\mu _ p$, and hence $m_ u$, is the trace measure of $\alpha +\mathcal Du$ and does not depend on $q$.

We also use two elementary facts. For a nonnegative Hermitian matrix $P$ and any matrix $M$ we have $|\sum _{l,m}p_{l\bar m}M_{lm}|\leq \| M\| \operatorname {tr}P$, by diagonalizing $P$; in particular $|P(y,\bar y')|\leq |y||y'|\operatorname {tr}P$. Moreover, since $a(0,\rho )$ is the Taylor polynomial of degree $N$ of $F(0,\cdot )$, which is that of $J_0$, we have $D_\rho a=I+O(r^2)$ and $\partial _ za(0,ws)=I+O(r)$, and $a(0,ws)=ws+O(r^3)$, for $|s|\leq 2$. Hence the map $s\mapsto a(0,ws)$ is injective on $\{ |s|\leq 2\} $, has real Jacobian determinant at least $r^ d/2$, and takes values in $B(0,3r)$.

We first bound the masses of $p$ on small balls about the center; this part does not use the smoothness of $p$. By the Jensen–Lelong formula (theorem 6.7.6), the normalized trace mass $D_ p(x,t)$ of (6.7.5) is nondecreasing in $t$, and $\mu _ p(B(x,t))$ is a fixed multiple of $t^{d-2}D_ p(x,t)$. On a fixed larger ball the trace mass of $p$ is bounded by $C\| p\| _{L^1}$ of a still larger ball: pair $\Delta p\geq 0$ with a cutoff function. Hence

For the second estimate, (6.7.7) and the bound for $D_ p$ give $M_ p(x,\varrho )\geq M_ p(x,R)-2C\log (R/\varrho )$ for $\varrho <R$, where $M_ p$ denotes spherical means. Averaging over $R$ in a fixed interval bounds the right side below by $-C(1+|\log \varrho |)$, since the average of $|p|$ over a fixed annulus is bounded in terms of $\| u\| _{L^1}$. As $|p|=2p^+-p$ and $p^+$ is bounded in terms of $\sup _ Xu$, integration over spheres gives $\int _{B(x,t)}|p|\leq Ct^ d(1+|\log t|)$, and $|u|\leq |p|+|q|$. In particular $m_ u\leq Cr^{-2}$.

To bound $E$, note that $|Y|\leq C(|v|+|\eta |)$ and $P[Y]^2\leq |Y|^2\operatorname {tr}P$ for $|s|\leq 2$. Moreover $\mathcal LG=\sum _ mb_ m\partial _{s_ m}G$ with $b=w^{-1}(D_\rho a)^{-1}Y$, by (16.17.12) and its proof, and $|b|+|\nabla _ sb|\leq Cr^{-1}(|v|+|\eta |)$; an integration by parts gives $\int \mathcal LG\, \delta _1\, \mathrm ds=-\int G\sum _ m\partial _{s_ m}(b_ m\delta _1)\, \mathrm ds$. By (16.17.13),

Changing variables by the map $s\mapsto a(0,ws)$ and using the two mass bounds, we obtain $|E|\leq Cr^{N-2}(1+|\log r|)(|v|^2+|\eta |^2)\leq Cr^2(|v|^2+|\eta |^2)$, since $N=8$.

Now we bound the right side of (16.17.16) from below. Its first integral is nonnegative. The formulas for the coefficients give $|\Xi |\leq C|v|$ and $|\Theta |\leq (C_ d+Cr)|v|$, where $C_ d$ is the norm of the linear map $v\mapsto \Theta $ formed by the terms with $|I|=2$, which do not depend on $s$ and $w$; the terms with $|I|\geq 3$ carry a factor $\bar w$. Every term of $U$ except the last carries the factor $-\chi _1/\chi =(1-|s|^2)^2\in [0,1]$; after this factor is removed, the curvature term has norm $\| C^ v\| \leq C_ c|v|^2$, where $C_ c$ depends only on the $c_{jklm}$, the $e$ term at most $Cr|v|^2$ and the mixed $d$ term at most $C|v||\eta |$, while the last term $\beta \beta ^*$ of $U$ gives the nonnegative contribution $r^2\int P[D_\rho a\, \beta ]^2\chi \, \mathrm ds$. Put

The substitution $s\mapsto w^{-1}a(0,ws)=s+O(r^2)$, whose derivative is $I+O(r^2)$, changes the kernel $-\chi _1(|s|^2)$ by $O(r^2)$ on $\{ |s|\leq 2\} $, and the unitary substitution $s\mapsto (w/r)s$ does not change it. Hence $r^2\int (-\chi _1)|D_\rho a|^2\operatorname {tr}P\, \mathrm ds\leq \frac12\lambda _0+Cr^4m_0$, and similarly $r^2\int _{|s|\leq 1}\operatorname {tr}P\, \mathrm ds\leq Cr^2m_0$. Consequently the second and third integrals of (16.17.16), the second including the term $-r^2\Xi \Xi ^*$ from the square, are bounded below by

Thus $A$ depends only on the fixed geometry, and since also $|E|\leq Cr^2(|v|^2+|\eta |^2)$, we obtain

For $v=0$ the coefficients $\Xi $, $\Theta $ and $U$ vanish, and (16.17.16) reads $\partial _ w\partial _{\bar w}\Psi _ p=\int P[D_\rho a\, s]^2\chi \, \mathrm ds+E$.

It remains to subtract the smoothing of the smooth function $q$. Since $q$ is smooth, $\Psi _ q(z,w)$ is smooth also at $w=0$, where it equals $q(z)$. By rotation invariance, the terms of degree one and the terms in $w^2$ and $\bar w^2$ of its Taylor expansion in $w$ vanish, so $\Psi _ q(z,w)-q(z)$ is $|w|^2$ times a smooth function of $z$, up to an error $O(|w|^3)$ whose derivatives lose one power of $|w|$ for each derivative in $w$. Hence, with $\mathcal Dq=\alpha $,

Since $\Psi _ u=\Psi _ p-\Psi _ q$ and $\partial _ w\partial _{\bar w}(K|w|^2)=K$, we choose $K\geq C_ q+1$. Adding these estimates proves (16.17.2) with $\lambda _0$ in place of $\lambda _ u$ and $m_0$ in place of $m_ u$. Moreover, for $v=0$ it gives $\partial _ w\partial _{\bar w}H_ u\geq K-C_ q-Cr^2>0$. Since $\partial _ w\partial _{\bar w}g=\frac1{4r}\partial _ r(r\partial _ rg)$ for a function $g$ of $r=|w|$, the second derivative of $H_ u$ with respect to $\log r$ is $(r\partial _ r)^2H_ u=4r^2\partial _ w\partial _{\bar w}H_ u>0$, so $H_ u$ is a strictly convex function of $\log r$.

Finally we compare $\lambda _0$ with $\lambda _ u$. Let $\Psi ^{\mathrm e}_ g(r)=\int g(x+rs)\chi (|s|^2)\, \mathrm ds$ denote the Euclidean radial convolution in the normal coordinates. For a smooth function $g$, the radial derivative is

Indeed, $r\partial _ r[g(x+rs)]=\sum _ m\bigl (s_ m\partial _{s_ m}+\bar s_ m\partial _{\bar s_ m}\bigr )[g(x+rs)]$, and by $s_ m\chi =\partial _{\bar s_ m}\chi _1$ and an integration by parts, $\int s_ m\partial _{s_ m}[g(x+rs)]\chi \, \mathrm ds=-\int \partial _{s_ m}\partial _{\bar s_ m}[g(x+rs)]\chi _1\, \mathrm ds=-r^2\int g_{m\bar m}(x+rs)\chi _1\, \mathrm ds$; the terms with $\bar s_ m$ are the complex conjugates. In particular $r\partial _ r\Psi ^{\mathrm e}_ p=\lambda _0$, and $|r\partial _ r\Psi ^{\mathrm e}_ q|\leq Cr^2$.

By (16.17.8), the substitution $s\mapsto r^{-1}F(x,rs)=s+O(r^2)$, with derivative $I+O(r^2)$, shows that the kernel of $\Psi _ u(x,r)$ differs from that of $\Psi ^{\mathrm e}_ u(r)$ by at most $Cr^{2-d}$ on $B(x,2r)$, where $\Psi ^{\mathrm e}_ u=\Psi ^{\mathrm e}_ p-\Psi ^{\mathrm e}_ q$. Hence the radial function $\psi (r)=\Psi _ u(x,r)-\Psi ^{\mathrm e}_ u(r)$ satisfies $|\psi (r)|\leq Cr^{2-d}\int _{B(x,2r)}|u|\leq Cr^2(1+|\log r|)$, so $\psi (r)\to 0$ as $r\to 0$; like the mass bounds, this holds for every $u\in \mathrm{PSH}(X,\theta )$. In the same way, by the case $v=0$ of (16.17.16), the matrix-valued kernels against which $P$ is integrated in $\partial _ w\partial _{\bar w}\Psi _ p$ and in $\partial _ w\partial _{\bar w}\Psi ^{\mathrm e}_ p=\int P(x+ws)[s]^2\chi \, \mathrm ds$ differ in norm by at most $Cr^{2-d}$ on $B(x,3r)$; since $|P[M]|\leq \| M\| \operatorname {tr}P$, these vertical derivatives differ by at most $Cr^2m_0+|E|\leq C$. The smooth function $q$ contributes bounded vertical derivatives as well. Thus $|\partial _ r(r\partial _ r\psi )|=4r|\partial _ w\partial _{\bar w}\psi |\leq Cr$ for all small $r$.

Integrating from $r'$ to $r$ shows that $\ell =\lim _{r'\to 0}r'\psi '(r')$ exists and $|r\psi '(r)-\ell |\leq Cr^2$. If $\ell \neq 0$, then $\psi '(r)=\ell /r+O(r)$, and $\psi (r)-\psi (r')=\ell \log (r/r')+O(r^2)$ would be unbounded as $r'\to 0$, contradicting $\psi \to 0$. Hence $\ell =0$ and $|r\partial _ r\psi |\leq Cr^2$. Since $\lambda _ u=r\partial _ r\Psi ^{\mathrm e}_ p-r\partial _ r\Psi ^{\mathrm e}_ q+r\partial _ r\psi +2Kr^2$, we obtain $|\lambda _ u-\lambda _0|\leq Cr^2$. Since $m_0\leq m_ u$, this proves (16.17.2) for smooth $p$; we record that it holds with $m_0$ in place of $m_ u$.

Step 6: nonsmooth potentials. Let $p$ be plurisubharmonic near $\overline{B(0,4r)}$. We approximate $p$ by its mollifications $p_\varepsilon $, $0<\varepsilon <r$ (section 2.5), and pass to the limit in the estimates of Step 5. By (3.4.13), applied to the subharmonic function $p$, we have $p\leq p_\varepsilon \leq \sup _{B(0,4r)}p$, so $|p_\varepsilon |\leq |p|+C$ on $B(0,3r)$. The functions $p_\varepsilon $ are smooth and plurisubharmonic on $B(0,3r)$, by proposition 5.9.2(2). Since derivatives commute with mollification (proposition 2.5.3(1)), the trace measure of $p_\varepsilon $ is the mollification of $\mu _ p$, so $\mu _{p_\varepsilon }(B(0,3r))\leq \mu _ p(B(0,4r))$, and the monotonicity of the normalized trace mass of $p_\varepsilon $ gives $\mu _{p_\varepsilon }(B(0,t))\leq Ct^{d-2}$ for $t\leq 3r$. Thus the mass bounds used above hold for $p_\varepsilon $ on these balls, with constants independent of $\varepsilon $, and (16.17.2) holds for $u_\varepsilon =p_\varepsilon -q$ at the center, with $1+r^{-d}\mu _{p_\varepsilon }(B(0,3r))\leq m_ u$ in place of $m_ u$; so do the inequalities $|\lambda _{u_\varepsilon }-\lambda _0(p_\varepsilon )|\leq Cr^2$ and $\partial _ w\partial _{\bar w}H_{u_\varepsilon }\geq K-C_ q-Cr^2$.

Since $p_\varepsilon \to p$ in $L^1(B(0,3r))$ (lemma 2.5.1(2)), and $\Psi $, its derivatives in $(z,w)$ for $w\neq 0$ and $\Psi ^{\mathrm e}$ are given by integration against smooth kernels, the joint Levi form of $H_{u_\varepsilon }$ at $(0,w)$, $\lambda _{u_\varepsilon }$ and $\lambda _0(p_\varepsilon )=r\partial _ r\Psi ^{\mathrm e}_{p_\varepsilon }$ converge to the corresponding quantities for $u$. Letting $\varepsilon \to 0$ proves (16.17.2) and the strict convexity of $H_ u$ in $\log r$ for every $u\in \mathrm{PSH}(X,\theta )$. Finally, inserting $m_ u\leq Cr^{-2}$, or the hypothesis $m_ u\leq M(x)/r$, and using $r<1$, we obtain (16.17.3) and (16.17.4).

Step 7: the remaining assertions. We first study $H_ u(x,r)$ as $r\to 0$; by Steps 5 and 6 it is a convex function of $\log r$. The kernel of $\Psi _ u$ is positive with mass one, so the upper bound for $u$ bounds $H_ u(x,r)$ from above as $r\to 0$. A convex function of $\log r$ that is bounded above as $r\to 0$ is nondecreasing: if its slope were negative at some point, the tangent line there, which lies below the function, would tend to $+\infty $ as $r\to 0$. Hence its slope $\lambda _ u=r\partial _ rH_ u$ is nonnegative.

To identify the limit, recall from theorem 3.4.5(2), applied to the subharmonic function $p$ (theorem 5.9.1), that the spherical means of $p$ about $x$ decrease to $p(x)$ as the radius decreases to zero; since $\Psi ^{\mathrm e}_ p(r)$ is an average of these spherical means at radii at most $r$, monotone convergence gives $\Psi ^{\mathrm e}_ p(r)\to p(x)$, and $\Psi ^{\mathrm e}_ q(r)\to q(x)$. Since $\psi (r)\to 0$, we get $H_ u(x,r)\to u(x)$ as $r\to 0$. If $u$ is bounded near $x$, this limit is finite. The slope of a convex function is nondecreasing, so it has a limit as $r\to 0$; a positive limit would force $H_ u(x,r)\to -\infty $. Therefore $\lambda _ u(x,r)\to 0$.

Next we prove the two comparison statements. If $u$ is bounded on $B(x,2r)$, the comparison of the kernels in Step 5 gives $|\Psi _ u(x,r)-\Psi ^{\mathrm e}_ u(r)|\leq Cr^2\sup _{B(x,2r)}|u|$, which is the asserted comparison with radial convolution. Now let $f$ be a $C^1$ function whose weak complex Hessian is bounded by $M_ f$. In normal coordinates its Euclidean Laplacian $4\sum _ mf_{m\bar m}$ is bounded by $CM_ f$, and so is the Laplacian of its mollifications $f_\varepsilon $. By (16.17.18), $|\partial _ r\Psi ^{\mathrm e}_{f_\varepsilon }(r)|\leq CM_ fr$, and since $\Psi ^{\mathrm e}_{f_\varepsilon }(r)\to f_\varepsilon (x)$ as $r\to 0$, we get $|\Psi ^{\mathrm e}_{f_\varepsilon }(r)-f_\varepsilon (x)|\leq CM_ fr^2$. Letting $\varepsilon \to 0$, the same bound holds for $f$, by uniform convergence. Finally, by (16.17.8), $|f(F(x,r\zeta ))-f(x+r\zeta )|\leq Cr^3\| \nabla f\| _\infty $ for $|\zeta |\leq 1$, and $h(0)=I$, so $|\Psi _ f(x,r)-\Psi ^{\mathrm e}_ f(r)|\leq Cr^3\| \nabla f\| _\infty $. This proves (16.17.5).

Finally we prove (16.17.6). We work in a fixed coordinate chart with a smooth frame of $T X$ that is unitary for $h$, obtained by the Gram–Schmidt process, and we describe a tangent vector by its vector $\zeta \in \mathbb C^ n$ of coefficients in this frame. Then $\mathrm d\zeta _ h=\mathrm d\zeta $, and the frame matrix $S(x)$, which satisfies $SS^*=(h^{i\bar j})$, gives the vector $S(x)\zeta $ of coordinate components. In the chart, (16.17.7) shows that $F(x,\zeta )$ differs by $O(|\zeta |^{N+1})$ from $J_ z(S(x)\zeta )$, where $z$ is the coordinate of $x$ and $J_ z(\xi )=z+\xi -\frac12\Gamma (z)(\xi ,\xi )+O(|\xi |^3)$ is the jet map of the chart. Hence $F(x,\zeta )=x+S(x)\zeta +Q_ x(S(x)\zeta ,S(x)\zeta )+O(|\zeta |^3)$ with the holomorphic quadratic map $Q_ x=-\frac12\Gamma (z)$.

The first moments of the radial weight vanish, and so do the moments $\int \zeta _ i\zeta _ j\chi (|\zeta |^2)\, \mathrm d\zeta $, by the substitution $\zeta \mapsto \mathrm e^{\mathrm it}\zeta $; by symmetry, $\int \zeta _ i\bar\zeta _ j\chi (|\zeta |^2)\, \mathrm d\zeta =a_\chi \delta _{ij}$. For smooth $g$, the second-order Taylor expansion of $g(F(x,r\zeta ))$ in $r\zeta $, integrated against $\chi (|\zeta |^2)\, \mathrm d\zeta $, therefore gives

uniformly on compact subsets of the chart.

For the distributional statement let $\varphi $ be a test function in the chart. For fixed $\zeta $ near $0$ the map $\Phi _\zeta (x)=F(x,\zeta )$ is a diffeomorphism near $\operatorname{Supp}\varphi $, with $\Phi _0$ the identity, and its inverse depends smoothly on $(\zeta ,y)$, by the inverse function theorem. By Fubini's theorem and the change of variables $y=\Phi _{r\zeta }(x)$, the adjoint kernel is

The integrand is a smooth function of $(r\zeta ,y)$, equal to $\varphi (y)$ at $r=0$, so Taylor's formula in $r\zeta $ gives $\Psi _ r^*\varphi =\varphi +r\varphi _1+r^2\varphi _2+O(r^3)$, with continuous functions $\varphi _1,\varphi _2$ supported in a fixed compact set and a remainder bounded uniformly in $y$. Pairing this with an arbitrary smooth $g$ and comparing with the expansion of $\Psi _ g$, we find $\int g\varphi _1=0$ and $\int g\varphi _2=a_\chi \int \varphi \sum h^{i\bar j}g_{i\bar j}$ for every smooth $g$; hence $\varphi _1=0$ and $\varphi _2=a_\chi \sum _{i,j}\partial _ i\partial _{\bar j}(h^{i\bar j}\varphi )$. Pairing the expansion of $\Psi _ r^*\varphi $ with a locally integrable $u$ gives

which is (16.17.6), because the right side is $a_\chi \langle D_ hu,\varphi \rangle $ by the definition of distributional derivatives. $\square$