16.15 Penalized equations and metric-Laplacian envelopesF9CE743055E9

Proof. The form is closed if and only if its $(2,1)$-part $\frac{\mathrm i}{2\pi }\partial _ le_{i\bar j}\, \mathrm dz_ l\wedge \mathrm dz_ i\wedge \mathrm d\bar z_ j$ and the conjugate $(1,2)$-part vanish, that is, if and only if $\partial _ le_{i\bar j}=\partial _ ie_{l\bar j}$; this holds in all holomorphic coordinates. If $z=\Phi (w)$ with $\Phi $ holomorphic and $J=D\Phi $ its complex Jacobian matrix, then $\mathrm dz_ i=J_{ia}\, \mathrm dw_ a$, so the new coefficients are $\tilde e(w)=J(w)^{\mathrm T}e(\Phi (w))\overline{J(w)}$.

Let $A=\overline{e(z_0)^{-1/2}}$, where $e(z_0)^{-1/2}$ is the positive square root of the inverse matrix; its operator norm is $\lambda (z_0)^{-1/2}$, where $\lambda (z)=\min _{|\xi |=1}\sum _{i,j}e_{i\bar j}(z)\xi _ i\bar\xi _ j$ is the smallest eigenvalue of $e(z)$, and $A^{\mathrm T}e(z_0)\bar A=e(z_0)^{-1/2}e(z_0)e(z_0)^{-1/2}=I$. In the coordinates $v$ with $z=z_0+Av$ the coefficient matrix $e'$ satisfies $e'(0)=I$. Put $c^ m_{ij}=\partial _{v_ i}e'_{j\bar m}(0)$, which is symmetric in $i$ and $j$ by the closedness, and define $v=w-\frac12c(w,w)$, that is, $v_ m=w_ m-\frac12c^ m_{ij}w_ iw_ j$. Then $J_{ia}=\partial v_ i/\partial w_ a=\delta _{ia}-c^ i_{aj}w_ j$, so $J(0)=I$ and $\partial _{w_ p}J_{ia}(0)=-c^ i_{ap}$. Differentiating $\tilde e_{a\bar b}=J_{ia}e'_{i\bar j}(v(w))\overline{J_{jb}}$ at $w=0$, and using that $\overline{J_{jb}}$ and $\bar v$ are antiholomorphic in $w$, we find

by the closedness, and $\partial _{\bar w_ p}\tilde e_{a\bar b}(0)=\overline{\partial _{w_ p}\tilde e_{b\bar a}(0)}=0$. Put $\Phi _{z_0}(w)=z_0+A(w-\frac12c(w,w))$. Its derivative at $0$ is the invertible matrix $A$, so it is a biholomorphism near $0$ by the holomorphic inverse function theorem (proposition 5.2.4(1)).

By the chain rule, the derivatives of $\tilde e$ at $0$ are polynomials in the derivatives of $e$ at $z_0$ and in the entries of $A$, $\bar A$ and $c$, and the entries of $c$ are polynomials in the first derivatives of $e$ at $z_0$ and in the entries of $A$ and $\bar A$. The derivatives of $e$ are bounded on compact sets, and so are the entries of $A$, because the function $\lambda $ is continuous and positive, as a minimum over the unit sphere of a function that is continuous in $(z,\xi )$; hence these polynomials are bounded for $z_0\in K$. Finally, if $Q$ is unitary, replacing $\Phi _{z_0}(w)$ by $\Phi _{z_0}(Qw)$ replaces $\tilde e(w)$ by $Q^{\mathrm T}\tilde e(Qw)\bar Q$, which equals $Q^{\mathrm T}\bar Q=\overline{Q^*Q}=I$ at $0$, still has vanishing differential at $0$, and has derivatives at $0$ bounded by a constant depending only on $n$ times those of $\tilde e$. $\square$

Proof. At every point the eigenvalues of $g$ relative to $h$ are positive, their sum is $H$ and the sum of their inverses is $\operatorname {tr}_ gh$, so the Cauchy–Schwarz inequality gives $H\operatorname {tr}_ gh\geq n^2$.

Fix finitely many charts and compact subsets of their domains whose union is $X$, and let $x\in X$. By lemma 16.15.3, applied to $h$ in one of these charts, followed by a unitary change of coordinates that diagonalizes $g$ at $x$, there are coordinates as in the statement. In them, the derivatives of order at most two of $h$ at $x$ are bounded by a constant depending only on $X$ and $\omega $. All the following computations are at $x$, in these coordinates. Locally $\kappa $ and $\kappa +\mathrm{dd}^{\mathrm{c}}a$ have Kähler potentials (lemma 6.14.2): $g_{i\bar j}=\partial _ i\partial _{\bar j}\varphi $ with $\varphi $ smooth. Hence the derivatives $\partial _ pg_{i\bar q}$ are symmetric in $p$ and $i$, and the second derivatives $\partial _ p\partial _{\bar l}g_{i\bar q}$ are symmetric in $p,i$ and in $\bar l,\bar q$. Differentiating $H=h^{i\bar q}g_{i\bar q}$ twice, and using that the first derivatives of $h$, hence of $h^{-1}$, vanish at $x$, we get

Since $g$ is diagonal with positive eigenvalues, $|E|\leq C_ EH\operatorname {tr}_ gh$, where $C_ E$ bounds the second derivatives of $h^{-1}$ at $x$. As $h=I$ and $\mathrm dh=0$ at $x$, the formula $\partial (h^{-1})=-h^{-1}(\partial h)h^{-1}$ shows that these are the second derivatives of $-h$, up to transposition; so $C_ E$ depends only on $X$ and $\omega $.

On the other hand, differentiating the identity $\partial _ p\log \det g=g^{i\bar q}\partial _ pg_{i\bar q}$ and using $\partial (g^{-1})=-g^{-1}(\partial g)g^{-1}$ gives

By the symmetry of the second derivatives, $\partial _ p\partial _{\bar p}g_{i\bar i}=\partial _ i\partial _{\bar i}g_{p\bar p}$. Summing over $p$, comparing with the formula for $D_ gH$, and noting that $\sum _ p\partial _ p\partial _{\bar p}=D_ h$ at $x$, we obtain the identity (16.15.5), with the bound for $E$ just proved.

For the inequality for $\log H$, note that $\partial _ pH=\sum _ i\partial _ pg_{i\bar i}=\sum _ i\partial _ ig_{p\bar i}$ by the symmetry of the first derivatives. The Cauchy–Schwarz inequality with weights $\lambda _ i$ therefore gives

the last sum containing the previous one (take $p=q$ in the term of index $(i,p,q)$ and rename). Since $D_ g\log H=H^{-1}D_ gH-H^{-2}|\partial H|_ g^2$, the quadratic term in $D_ gH$ absorbs the negative term, and (16.15.6) follows. $\square$

Proof. Step 1: the value bounds. Since $(\kappa +\mathrm{dd}^{\mathrm{c}}a)^ n/\kappa ^ n=\det g/\det k$, the equation says that $\det g=\mathrm e^{\beta (a-f)}\det k$. At a maximum point of $a$ the complex Hessian $(a_{i\bar j})$ is negative semidefinite, so $0<g\leq k$ and therefore $\det g\leq \det k$ there, the eigenvalues of $k^{-1/2}gk^{-1/2}$ lying in $(0,1]$. Hence $a\leq f\leq \max _ Xf$ at that point, and therefore everywhere. At a minimum point of $a$ the inequalities are reversed, and $a\geq \min _ Xf$.

Step 2: a differential inequality for $\log H$. Let $x\in X$, and take the coordinates of lemma 16.15.4 at $x$. By that lemma, (16.15.6) holds at $x$, with $|E|\leq C_ EH\operatorname {tr}_ gh$, and $H\operatorname {tr}_ gh\geq n^2$. Now we use the equation in the form $\log \det g=\log \det k+\beta (a-f)$, valid in every chart. Since $D_ ha=H-\operatorname {tr}_ hk$,

The function $D_ h\log \det k$ is a smooth function on $X$, so it is bounded by a constant depending only on $X$, $\omega $ and $\kappa $. Dividing by $H$ and using $H^{-1}\leq \operatorname {tr}_ gh/n^2$ and $|E|/H\leq C_ E\operatorname {tr}_ gh$, we arrive at

where $C_0$ depends only on $X$, $\omega $ and $\kappa $. Both sides are independent of the coordinates, so (16.15.9) holds everywhere on $X$.

Step 3: the maximum principle. By (16.15.1) we have $B>0$. Let $B'\geq B$ be a constant, so that $\operatorname {tr}_ hk+D_ hf\leq B'$ on $X$. For a constant $A>0$, to be chosen, let $x_0$ be a maximum point of $Q=\log H-Aa$. Since

inequality (16.15.9) gives at $x_0$, where the complex Hessian of $Q$ is negative semidefinite,

We first prove the uniform equivalence of $g$ and $h$. Let $\beta $ be fixed, let $M\geq \| f\| _\infty $ and $N\geq \sup _ XD_ hf$ be given, take $B'=\sup _ X\operatorname {tr}_ hk+N$, and $A=(C_0+\beta B'/n^2+1)/\gamma $. Since $1/H\leq \operatorname {tr}_ gh/n^2$, the right side of (16.15.10) is at least $\beta +\operatorname {tr}_ gh-An$, so $\operatorname {tr}_ gh\leq An$ at $x_0$. Thus every eigenvalue of $g$ relative to $h$ is at least $(An)^{-1}$ at $x_0$. By Step 1, $|a-f|\leq \operatorname {osc}_ Xf\leq 2M$, and therefore, by the equation, the product of the eigenvalues satisfies

everywhere on $X$. Together with the lower bound for the eigenvalues at $x_0$, the upper bound for their product bounds each eigenvalue above at $x_0$, and hence bounds $H(x_0)$. For every $x\in X$, since $Q(x)\leq Q(x_0)$ and $\min _ Xf\leq a\leq \max _ Xf$,

Thus $H$ is bounded on $X$, so every eigenvalue of $g$ relative to $h$ is bounded above on $X$, and the lower bound for their product then bounds every eigenvalue below. All the constants depend only on $X$, $\omega $, $\kappa $, $\beta $, $M$ and $N$, which proves the uniform equivalence of $g$ and $h$.

Next we prove the bound for $H$ that is uniform in $\beta $. Take $B'=B$, $A=(C_0+1)/\gamma $ and $\beta _0=2An$; both $A$ and $\beta _0$ depend only on $X$, $\omega $ and $\kappa $. Suppose that $\beta >\beta _0$, and let $x_0$ be a maximum point of $Q=\log H-Aa$ for this $A$. If $H(x_0)>2B$, then $1-B/H>1/2$ at $x_0$, and the right side of (16.15.10) is at least $\beta /2+\operatorname {tr}_ gh-An>0$, a contradiction. Hence $H(x_0)\leq 2B$, and the same argument as above gives $H\leq 2B\, \mathrm e^{A\operatorname {osc}_ Xf}$ on $X$. Since $\operatorname {osc}_ Xf\leq 2\| f\| _\infty $ and $B\leq \sup _ X\operatorname {tr}_ hk+N$, this proves the last assertion. $\square$

Proof. (1) In the coordinates $w$ the metric and its inverse have the components

By these rules and the chain rule $\partial _{w_ c}(F\circ z)=J_{kc}\, (\partial _{z_ k}F)\circ z$, which holds because $\partial _{w_ c}\bar z_ k=0$, we have $\partial _{w_ c}g'_{a\bar b}=(\partial _{w_ c}J_{ia})g_{i\bar j}\overline{J_{jb}}+J_{ia}J_{kc}\, \partial _ kg_{i\bar j}\, \overline{J_{jb}}$, and hence

Let $T$ be a tensor, and differentiate its transformation rule by $\partial _{w_ c}$. By the product rule, this gives the transformed array $J_{kc}\partial _ kT$, plus one term for each holomorphic index, in which $\partial _{w_ c}$ falls on the Jacobian factor of that index; the factors of the antiholomorphic indices are antiholomorphic and are not differentiated. For an upper holomorphic index, the term contains $\partial _{w_ c}(J^{-1})_{dm}=-(J^{-1})_{dm'}(\partial _{w_ c}J_{m'b})(J^{-1})_{bm}$, and it cancels against the contribution of the second term of (16.15.12) to $\Gamma '^ d_{cb}T'^{\cdots b\cdots }$; for a lower holomorphic index $a$, the term contains $\partial _{w_ c}J_{ia}$, and it cancels against the contribution of the second term of (16.15.12) to $-\Gamma '^ b_{ca}T'_{\cdots b\cdots }$. The contributions of the first term of (16.15.12) are the transforms of the connection terms of $T$. Hence $\nabla _ cT'$ is the transform of $\nabla _ kT$, contracted with $J_{kc}$, so $\nabla _ pT$ is a tensor; by conjugation, so is $\nabla _{\bar q}T$.

The Leibniz rule follows from the product rule for $\partial _ p$ and the additivity of the connection terms over the indices. If an upper and a lower holomorphic index of $T$ are contracted, their connection terms $\Gamma ^ m_{pa}T^{a\cdots }{}_{\cdots m}$ and $-\Gamma ^ a_{pm}T^{m\cdots }{}_{\cdots a}$ cancel, so $\nabla $ commutes with contractions. Next,

because $g^{a\bar l}g_{a\bar j}=\delta _{lj}$, and $\nabla _{\bar q}g_{i\bar j}=\overline{\nabla _ qg_{j\bar i}}=0$, since $\overline{g_{i\bar j}}=g_{j\bar i}$. The connection terms of the identity tensor cancel, so $\nabla \delta =0$, and the Leibniz rule applied to $g^{i\bar j}g_{l\bar j}=\delta ^ i_ l$ gives $(\nabla g^{i\bar j})g_{l\bar j}=0$, hence $\nabla g^{-1}=0$. For a complete contraction $S$, a function, $\nabla _ pS=\partial _ pS$, and the last assertion follows from the Leibniz rule and the commutation with contractions.

(2) By (16.15.12) for $g$ and for $h$, the inhomogeneous terms of $\Gamma '$ and $\widehat\Gamma '$ coincide, so the difference transforms as a tensor. Moreover $\widehat\nabla _ ig_{j\bar l}=\partial _ ig_{j\bar l}-\widehat\Gamma ^ p_{ij}g_{p\bar l}$, and contracting with $g^{m\bar l}$ gives $\Gamma ^ m_{ij}-\widehat\Gamma ^ m_{ij}$.

(3) The second term of (16.15.12) is holomorphic in $w$, so it is annihilated by $\partial _{\bar w_ e}=\overline{J_{le}}\, \partial _{\bar z_ l}$, and $R$ transforms as a tensor; it is symmetric in $i$ and $j$ because $\Gamma $ is. Since $\Gamma ^ m_{im}=g^{m\bar l}\partial _ ig_{m\bar l}=\partial _ i\log \det g$, we get $R_{i\bar j}=-\partial _{\bar j}\partial _ i\log \det g$. Both sides of each of the two identities are tensors, so it suffices to prove them at a point $x$ in coordinates in which $g_{i\bar j}(x)=\delta _{ij}$ and $\mathrm dg(x)=0$, which exist by lemma 16.15.3. There $\Gamma =0$, so covariant derivatives at $x$ are ordinary derivatives, and $\partial g^{-1}=-g^{-1}(\partial g)g^{-1}$ and $\bar\partial g^{-1}$ vanish at $x$. Using a local Kähler potential $\varphi $, $g_{i\bar j}=\partial _ i\partial _{\bar j}\varphi $, whose derivatives commute, we find at $x$

and

in each case all terms containing a first derivative of $g$ or of $g^{-1}$ vanish at $x$.

(4) A tensor with only holomorphic indices receives no connection term from $\nabla _{\bar q}$, so $\nabla _{\bar q}T=\partial _{\bar q}T$. The tensor $\nabla _ pT$ again has only holomorphic indices, so $\nabla _{\bar q}\nabla _ pT^ m_{ij}$ is the $\partial _{\bar q}$-derivative of $\partial _ pT^ m_{ij}+\Gamma ^ m_{pa}T^ a_{ij}-\Gamma ^ a_{pi}T^ m_{aj}-\Gamma ^ a_{pj}T^ m_{ia}$. On the other hand, $\nabla _{\bar q}T=\partial _{\bar q}T$ has the additional antiholomorphic index $\bar q$, which receives no term from $\nabla _ p$, so $\nabla _ p\nabla _{\bar q}T^ m_{ij}=\partial _ p\partial _{\bar q}T^ m_{ij}+\Gamma ^ m_{pa}\partial _{\bar q}T^ a_{ij}-\Gamma ^ a_{pi}\partial _{\bar q}T^ m_{aj}-\Gamma ^ a_{pj}\partial _{\bar q}T^ m_{ia}$. The difference consists of the terms in which $\partial _{\bar q}$ falls on $\Gamma $; since $\partial _{\bar q}\Gamma ^ m_{pa}=-R^ m{}_{a\bar qp}$ by the symmetry of $R$, this is the asserted formula. $\square$

Proof. In this proof $C,C_1,C_2,\ldots $ denote constants depending only on $X$, $\omega $, $\kappa $, $\beta $, $C_0$ and $f$. We show that $D_ gS\geq -C_1(S+1)$ and $D_ gH\geq C_0^{-1}S-C_2$, where $H=\operatorname {tr}_ hg$, and apply the maximum principle to $S+KH$ for a large constant $K$.

By lemma 16.15.11(1),(2), $S$ is a smooth function on $X$. The fixed smooth tensors $\widehat R$ (the tensor of lemma 16.15.11(3) for $h$), $\widehat\nabla \widehat R$, $\tau =\operatorname {Ric}(k)+\beta k+\beta t(f_{i\bar j})$ and $\widehat\nabla \tau $ have $h$-norms bounded on $X$ independently of $t\in [0,1]$, because they are smooth on the compact manifold $X$ and depend affinely on $t$; by the assumption on $g$, their $g$-norms are bounded as well.

The Laplacian of $\Psi $. Let $R$ be the tensor of lemma 16.15.11(3) for $g$. Since $\Psi $ has only holomorphic indices,

Put $D_ gT=g^{p\bar q}\nabla _ p\nabla _{\bar q}T$ and $\overline D_ gT=g^{p\bar q}\nabla _{\bar q}\nabla _ pT$ for tensors $T$. On a function $u$ both operators reduce to $D_ gu=g^{p\bar q}\partial _ p\partial _{\bar q}u$, since $\nabla _ p$ adds no term to the antiholomorphic index of $\partial _{\bar q}u$, and $\nabla _{\bar q}$ none to the holomorphic index of $\partial _ pu$. Applying $g^{p\bar q}\nabla _ p$ to the last identity and using $\nabla g^{-1}=0$ and lemma 16.15.11(3), we obtain

By lemma 16.15.11(4) and the first identity of (3), $\overline D_ g\Psi =D_ g\Psi +\mathcal R\Psi $, where

The Laplacian of $S$. By lemma 16.15.11(1) and $\nabla _ r\overline{\Psi ^ l_{pq}}=\overline{\nabla _{\bar r}\Psi ^ l_{pq}}$, which holds because $\nabla _ r$ adds no term to antiholomorphic indices,

The right side is a tensor with one lower holomorphic index $r$, on which $\nabla _{\bar s}$ acts as $\partial _{\bar s}$. Applying lemma 16.15.11(1) once more and contracting with $g^{r\bar s}$, we obtain, with the inner products and norms defined by $g$,

where $\nabla \Psi =(\nabla _ r\Psi ^ m_{ij})$ and $\overline\nabla \Psi =(\nabla _{\bar s}\Psi ^ m_{ij})$; for the last term we used $\overline{g^{r\bar s}}=g^{s\bar r}$. Inserting $\overline D_ g\Psi =D_ g\Psi +\mathcal R\Psi $, we get

in particular $\langle \mathcal R\Psi ,\Psi \rangle _ g$ is real.

Bounds from the equation. In every chart, $\log \det g=\log \det k+\beta (a-tf)$, so

Hence $R_ i{}^ m=g^{m\bar l}\tau _{i\bar l}-\beta \delta ^ m_ i$ is bounded in $g$-norm, and so $|\langle \mathcal R\Psi ,\Psi \rangle _ g|\leq CS$. Moreover $\nabla _ iR_ j{}^ m=g^{m\bar l}\nabla _ i\tau _{j\bar l}$, because $\nabla g^{-1}=0$ and $\nabla \delta =0$. For a tensor the difference $\nabla _ p-\widehat\nabla _ p$ consists of the connection terms with $\Gamma -\widehat\Gamma =\Psi $ in place of $\Gamma $; thus $\nabla _ i\tau _{j\bar l}=\widehat\nabla _ i\tau _{j\bar l}-\Psi ^ a_{ij}\tau _{a\bar l}$ and

By (16.15.14) it follows that $|D_ g\Psi |_ g\leq C(1+S^{1/2})$. Discarding the two nonnegative gradient terms in (16.15.15), we conclude that

A compensating term. We use the trace $H=\operatorname {tr}_ hg$. Let $x\in X$, and take the coordinates of lemma 16.15.4 at $x$. By that lemma, identity (16.15.5) holds at $x$, with $|E|\leq C_ EH\operatorname {tr}_ gh\leq C$ by the assumption on $g$, and $D_ h\log \det g=D_ h\log \det k+\beta (H-\operatorname {tr}_ hk-tD_ hf)$ is bounded by $C$. At $x$ the coefficients $\widehat\Gamma $ vanish, so $\Psi ^ m_{ij}=\lambda _ m^{-1}\partial _ ig_{j\bar m}$ and

Hence the quadratic sum in (16.15.5) is at least $(\min _ p\lambda _ p)S\geq C_0^{-1}S$, and $D_ gH\geq C_0^{-1}S-C_2$ at $x$. Both sides are functions on $X$, so this inequality holds everywhere.

The maximum principle. Choose $K$ with $KC_0^{-1}\geq C_1+1$, and let $x_0$ be a maximum point of $S+KH$. There the complex Hessian of $S+KH$ is negative semidefinite, so

Thus $S\leq C_1+KC_2$ at $x_0$. Since $H\leq nC_0$ on $X$, the function $S+KH$, and hence $S$, is bounded on $X$.

First derivatives of $g$. In a chart of the fixed atlas, $\partial _ pg_{i\bar q}=\widehat\nabla _ pg_{i\bar q}+\widehat\Gamma ^ l_{pi}g_{l\bar q}=g_{m\bar q}\Psi ^ m_{pi}+\widehat\Gamma ^ l_{pi}g_{l\bar q}$ by lemma 16.15.11(2). On a compact subset of the chart, $h$ lies between two positive multiples of the identity matrix and $\widehat\Gamma $ is bounded, so both terms are bounded, by the bounds for $S$ and $g$. Finally $\partial _{\bar p}g_{i\bar q}=\overline{\partial _ pg_{q\bar i}}$, so all first derivatives of the $g_{i\bar j}$ in the underlying real coordinates are bounded. $\square$

Proof. (1) Fix $s$ with $0<|s|<s_0$. We claim that $\delta _ sZ(x)=\int _0^1\partial _ eZ(x+tse)\, \mathrm dt$ for almost every $x\in D'$. For smooth $Z$ this is the fundamental theorem of calculus. In general, let $K\subset D'$ be compact; the set $K'=\{ x+tse:x\in K,\ 0\leq t\leq 1\} $ is a compact subset of $D$. The mollifications $Z_\varepsilon =Z*\rho _\varepsilon $ are smooth near $K'$ for small $\varepsilon $, they converge to $Z$ in $L^1(K')$, and $\partial _ eZ_\varepsilon =(\partial _ eZ)*\rho _\varepsilon \to \partial _ eZ$ in $L^1(K')$, by proposition 2.5.3(1) and lemma 2.5.1(2). Hence the identity for $Z_\varepsilon $ passes to the limit in $L^1(K)$; for the right side we use $\int _ K\int _0^1|F(x+tse)|\, \mathrm dt\, \mathrm dx\leq \int _{K'}|F|$, by Tonelli's theorem. This proves the claim. By Hölder's inequality on $[0,1]$ and Tonelli's theorem,

Similarly, since $\delta _ sZ(x)-\partial _ eZ(x)=\int _0^1\bigl (\partial _ eZ(x+tse)-\partial _ eZ(x)\bigr )\, \mathrm dt$ for almost every $x\in D'$, Hölder's inequality and Tonelli's theorem give, with $F$ the extension of $\partial _ eZ$ by zero to $\mathbb R^ d$,

which tends to zero as $s\to 0$ by the continuity of translation in $L^ p(\mathbb R^ d)$ (lemma 2.5.1(2)).

(2) Let $\chi \in C_ c^\infty (D')$ with $0\leq \chi \leq 1$, extend $\chi Z$ by zero, and let $0<|s|<\min \{ s_1,\operatorname {dist}(\operatorname{Supp}\chi ,\mathbb R^ d\setminus D')\} $. Then $\delta _ s(\chi Z)$ vanishes outside $D'$, and on $D'$ it equals $(\tau _ s\chi )\delta _ sZ+(\delta _ s\chi )Z$. Since $D'$ is bounded and $p\geq 2$, Hölder's inequality gives

The function $\chi Z$ lies in $L^1\cap L^2$, so its $L^2$ Fourier transform is given by the integral (theorem 2.9.4), and $\widehat{\delta _ s(\chi Z)}=m_ s\widehat{\chi Z}$ with $m_ s(\xi )=(\mathrm e^{2\pi \mathrm is\xi _ e}-1)/s$, where $\xi _ e$ is the component of $\xi $ in the direction $e$. By Plancherel's theorem, $\int |m_ s|^2|\widehat{\chi Z}|^2\leq M'^2$, and as $m_ s(\xi )\to 2\pi \mathrm i\xi _ e$, Fatou's lemma gives $\int 4\pi ^2\xi _ e^2|\widehat{\chi Z}|^2\leq M'^2$. Let $F\in L^2$ be the inverse $L^2$ transform of $2\pi \mathrm i\xi _ e\widehat{\chi Z}$. For a test function $\phi $ we have $\widehat{\partial _ e\phi }=2\pi \mathrm i\xi _ e\widehat\phi $, by integration by parts, so Parseval's identity, which extends to $L^2$ because the transform is unitary, gives

Thus $\partial _ e(\chi Z)=F\in L^2$. Since $\chi $ may equal $1$ near any given compact subset of $D'$, it follows that $\partial _ eZ\in L^2_{\mathrm{loc}}(D')$.

Now let $D''\Subset D'''\Subset D'$ be open. By (1), applied with $D'''$ in place of $D$ and $p=2$, we have $\delta _ sZ\to \partial _ eZ$ in $L^2(D'')$, so along a sequence $s_ j\to 0$ the functions $\delta _{s_ j}Z$ converge to $\partial _ eZ$ almost everywhere on $D''$. Fatou's lemma gives $\| \partial _ eZ\| _{L^ p(D'')}\leq \varliminf _ j\| \delta _{s_ j}Z\| _{L^ p(D'')}\leq M$. Exhausting $D'$ by such sets $D''$ and using monotone convergence, we obtain $\| \partial _ eZ\| _{L^ p(D')}\leq M$. $\square$

Proof. For the quantitative assertion, $C$ denotes constants depending only on the quantities listed there; $O'\Subset O\Subset O_1\Subset U$ denote open sets, and the constants may also depend on them. The qualitative assertion follows from the quantitative one. Indeed, if $V\Subset U'\Subset U$ with $U'$ open, then on $U'$ the matrix $G$ is bounded above and below and uniformly continuous, and $u$ has finite $C^1$ norm and bounded second derivatives. The quantitative assertion, applied on $U'$, therefore gives $u\in W^{\ell ,p}(V)$ for all $\ell $ and $p$, and Morrey's inequality (lemma 2.9.16), applied to the derivatives of $u$, shows that all derivatives of $u$ are continuous on $V$.

We prove the estimate by bootstrapping: an interior $L^ p$ estimate (Step 1), applied to difference quotients, raises the order of differentiability by one at a time (Steps 2 and 3), on smaller domains and with smaller exponents, and Step 4 chains these steps.

Step 1: an interior estimate. Let $(a^{i\bar j})$ be a function on $O$ with $\overline{a^{i\bar j}}=a^{j\bar i}$, $\Lambda ^{-1}|\zeta |^2\leq a^{i\bar j}\zeta _ i\bar\zeta _ j\leq \lambda ^{-1}|\zeta |^2$ and $|a(x)-a(y)|\leq \varpi '(|x-y|)$, where $\varpi '(r)\to 0$ as $r\to 0$. We claim that every real $w\in W^{2,p}_{\mathrm{loc}}(O)$ with $w\in W^{1,p}(O)$ satisfies

with $C$ depending also on $\varpi '$. Indeed, by (16.14.8), $a^{i\bar j}\partial _ i\partial _{\bar j}w=A:D^2w$ with a real symmetric $A\geq (4\Lambda )^{-1}I$, $|A|\leq C\lambda ^{-1}$ and $|A(x)-A(y)|\leq \varpi '(|x-y|)$. Let $\delta _0$ and $C_1$ be the constants of lemma 16.14.7 for $d=2n$, $p$ and $(4\Lambda )^{-1}$, and choose $r>0$ with $\varpi '(\varrho )\leq \delta _0$ for $0<\varrho <r$ and $r<\operatorname {dist}(O',\mathbb R^{2n}\setminus O)$. Cover $\overline{O'}$ by finitely many balls $B(x_\nu ,r/2)$ with $x_\nu \in \overline{O'}$, and let $\chi _\nu =\chi (\cdot -x_\nu )$, where $\chi \in C_ c^\infty (B(0,r))$ equals $1$ on $B(0,r/2)$. By lemma 2.9.14(1) and lemma 16.14.7,

and $A:D^2(\chi _\nu w)=\chi _\nu \, A:D^2w+2\sum _{\mu ,\sigma }A^{\mu \sigma }\partial _\mu \chi _\nu \, \partial _\sigma w+w\, A:D^2\chi _\nu $. Summing over $\nu $ proves the claim.

Step 2: $u\in W^{3,p}_{\mathrm{loc}}$. Let $e$ be a real coordinate unit vector, and let $0<|s|<\operatorname {dist}(O,\mathbb R^{2n}\setminus O_1)$, with the notation $\tau _ s$, $\delta _ s$ of lemma 16.15.16. For $x\in O$ and $0\leq t\leq 1$ the matrix $G_ t(x)=(1-t)G(x)+tG(x+se)$ satisfies $\lambda I\leq G_ t\leq \Lambda I$; let $(g_ t^{i\bar j})$ be its inverse and $a_ s^{i\bar j}=\int _0^1g_ t^{i\bar j}\, \mathrm dt$. By Jacobi's formula, $\frac{\mathrm d}{\mathrm dt}\log \det G_ t=g_ t^{i\bar j}(G_{i\bar j}(x+se)-G_{i\bar j}(x))$, so integration over $0\leq t\leq 1$ gives $\delta _ s\log \det G=a_ s^{i\bar j}\delta _ sG_{i\bar j}$. Since $\delta _ sG_{i\bar j}=\delta _ sk_{i\bar j}+\partial _ i\partial _{\bar j}\delta _ su$, equation (16.15.18) gives

The right side is bounded by $\sup _ U|\nabla \phi _0|+|\beta |\sup _ U|\nabla u|+C\lambda ^{-1}\sup _ U|\nabla k|$. The matrices $a_ s$ satisfy the hypotheses of Step 1 with $\varpi '=2\lambda ^{-2}\varpi $, because they are averages of inverses of matrices between $\lambda I$ and $\Lambda I$, and $|M^{-1}-N^{-1}|=|M^{-1}(N-M)N^{-1}|\leq \lambda ^{-2}|M-N|$ for such matrices. By lemma 16.15.16(1), $\| \delta _ su\| _{W^{1,p}(O)}\leq \| u\| _{W^{2,p}(O_1)}$. Step 1 therefore gives $\| D^2\delta _ su\| _{L^ p(O')}\leq C$, uniformly in $s$. Since $\delta _ s\partial _\mu \partial _\sigma u=\partial _\mu \partial _\sigma \delta _ su$, lemma 16.15.16(2), applied to $Z=\partial _\mu \partial _\sigma u$, shows that $\partial _ e\partial _\mu \partial _\sigma u\in L^ p(O')$, with norm at most $C$. Thus $\| u\| _{W^{3,p}(O')}\leq C$, where $C$ depends on $M$ through $\| u\| _{C^1(U)}$ and $\| D^2u\| _{L^ p(O_1)}$.

Step 3: from $W^{\ell }$ to $W^{\ell +1}$. Let $\ell \geq 3$ and suppose that $u\in W^{\ell ,q'}(O_1)$ for some $q'\geq 2p+2n$. By Morrey's inequality (lemma 2.9.16), applied to the derivatives of $u$ of order at most $\ell -2$, which lie in $W^{2,q'}(O_1)$, the function $u$ is of class $C^{\ell -1}$ on $O_1$, and on a slightly smaller set, which we again call $O_1$, its derivatives of order at most $\ell -1$ are bounded by $C\| u\| _{W^{\ell ,q'}}$. Thus $G\in C^{\ell -3}(O_1)$, its derivatives of order at most $\ell -3$ lie in $W^{1,2p}(O_1)$, and those of order $\ell -2$ lie in $L^{2p}(O_1)$.

We first record a formula for the derivatives of $\log \det G$. For a smooth function $G$ with values in the positive definite Hermitian matrices and a multi-index $\gamma $ with $|\gamma |\geq 1$,

where $P_\gamma (G)$ is a linear combination, with coefficients depending only on $\gamma $, of functions $\operatorname {tr}(G^{-1}\partial ^{\beta _1}G\, G^{-1}\partial ^{\beta _2}G\cdots G^{-1}\partial ^{\beta _ r}G)$ with $r\geq 2$, $|\beta _ i|\geq 1$ and $\beta _1+\cdots +\beta _ r=\gamma $; in particular $|\beta _ i|\leq |\gamma |-1$. For $|\gamma |=1$ this is Jacobi's formula, with $P_\gamma =0$. If it holds for $\gamma $, differentiate once more: by the Leibniz rule and $\partial _\mu (G^{-1})=-G^{-1}(\partial _\mu G)G^{-1}$, the derivative of each term $\operatorname {tr}(G^{-1}\partial ^{\beta _1}G\cdots )$ is a sum of terms of the same form, with $r$ or $r+1$ factors and total multi-index $\gamma +e_\mu $, and the derivative of $\operatorname {tr}(G^{-1}\partial ^\gamma G)$ is $\operatorname {tr}(G^{-1}\partial ^{\gamma +e_\mu }G)-\operatorname {tr}(G^{-1}\partial _\mu G\, G^{-1}\partial ^\gamma G)$. Note that $\operatorname {tr}(G^{-1}\partial ^\gamma G)=g^{i\bar j}\partial ^\gamma G_{i\bar j}$.

Now fix $\gamma $ with $|\gamma |=\ell -2\geq 1$. We claim that, almost everywhere on $O_1$,

where $\partial ^\gamma G$ is formed with weak derivatives. Let $K\subset O_1$ be compact, and let $u_\varepsilon =u*\rho _\varepsilon $ and $G_\varepsilon =k+(\partial _ i\partial _{\bar j}u_\varepsilon )$. By proposition 2.5.3(1) and lemma 2.5.1(1),(2), as $\varepsilon \to 0$ the derivatives of $u_\varepsilon $ of order at most $\ell -1$ converge uniformly on $K$ to those of $u$, and those of order $\ell $ converge in $L^{2p}(K)$. Hence $G_\varepsilon $ is positive definite on $K$ for small $\varepsilon $, its derivatives of order at most $\ell -3$ converge uniformly on $K$ to those of $G$, and $\partial ^\gamma G_\varepsilon \to \partial ^\gamma G$ in $L^{2p}(K)$. The formula above holds for the smooth functions $G_\varepsilon $ on the interior of $K$. Its left side converges to $\partial ^\gamma \log \det G=\partial ^\gamma \phi _0+\beta \partial ^\gamma u$ in the sense of distributions, since $\log \det G_\varepsilon \to \log \det G$ uniformly on $K$ and $\log \det G=\phi _0+\beta u$, and its right side converges in $L^1(K)$, since every factor of $P_\gamma (G_\varepsilon )$ involves derivatives of $G_\varepsilon $ of order at most $\ell -3$. Hence the identity holds in the sense of distributions on the interior of $K$, and therefore almost everywhere (proposition 2.5.3(4)). This proves the claim.

Put $v=\partial ^\gamma u\in W^{2,2p}(O_1)$. Since $\operatorname {tr}(G^{-1}\partial ^\gamma G)=D_ Gv+g^{i\bar j}\partial ^\gamma k_{i\bar j}$, the claim says that

For $x\in O$ and small $s$, as in Step 2, the discrete product rule $\delta _ s(FF')=(\tau _ sF)\delta _ sF'+(\delta _ sF)F'$ gives

We bound the right side in $L^ p(O)$ uniformly in $s$, using lemma 16.15.16(1) on $O\Subset O_1$. First, $\delta _ s(G^{-1})=-(\tau _ sG)^{-1}(\delta _ sG)G^{-1}$, so $\| \delta _ sG^{-1}\| _{L^{2p}(O)}\leq \lambda ^{-2}\| \partial _ eG\| _{L^{2p}(O_1)}\leq C$, as $\partial _ eG$ involves derivatives of $u$ of order three. Hence the last term is bounded by $\| \delta _ sG^{-1}\| _{L^{2p}}\| \partial \bar\partial v\| _{L^{2p}}\leq C$. Next, $\| \delta _ s\partial ^\gamma \phi _0\| _\infty \leq C$ and $\| \delta _ sv\| _{L^ p(O)}\leq \| \partial _ ev\| _{L^ p(O_1)}\leq C$. Finally, $g^{i\bar j}\partial ^\gamma k_{i\bar j}$ and $P_\gamma (G)$ are sums of products of bounded continuous factors, namely entries of $G^{-1}$, derivatives of $k$, and derivatives of $G$ of order at most $\ell -3$, each of which has difference quotients bounded in $L^ p(O)$; by the discrete product rule their difference quotients are bounded in $L^ p(O)$. Thus the right side is bounded in $L^ p(O)$, and $\| \delta _ sv\| _{W^{1,p}(O)}\leq \| \partial _ ev\| _{W^{1,p}(O_1)}\leq C$. The matrices $\tau _ sG^{-1}$ satisfy the hypotheses of Step 1 with $\varpi '=\lambda ^{-2}\varpi $. By Step 1, $\| D^2\delta _ sv\| _{L^ p(O')}\leq C$ uniformly in $s$, and lemma 16.15.16(2), applied to the second derivatives of $v$, shows that the third derivatives of $v$ lie in $L^ p(O')$, with norm at most $C$. Every multi-index of length $\ell +1$ is the sum of a multi-index of length $\ell -2$ and of three coordinate unit vectors, so $\| u\| _{W^{\ell +1,p}(O')}\leq C$, where $C$ depends on $u$ only through $\| u\| _{W^{\ell ,q'}(O_1)}$.

Step 4: conclusion. Choose open sets $V=O_{3\ell }\Subset O_{3\ell -1}\Subset \cdots \Subset O_0=U$, depending only on $V$, $U$ and $\ell $, and put $p_ m=2^{\ell -m}(p+2n)-2n$ for $3\leq m\leq \ell $. Then $p_\ell =p\geq 2$, $p_ m=2p_{m+1}+2n$, and $p_3\leq q$. Step 2, applied with the exponent $p_3$ in place of $p$ and with the sets $O_1$, $O_2$, $O_3$ in the roles of $O_1$, $O$, $O'$, bounds $\| u\| _{W^{3,p_3}(O_3)}$, since $\| D^2u\| _{L^{p_3}(O_1)}\leq |O_1|^{1/p_3-1/q}\| D^2u\| _{L^ q(U)}$. For $m=3,\ldots ,\ell -1$ we then apply Step 3 with $\ell $ replaced by $m$, with the exponent $p_{m+1}$ in place of $p$ and $q'=p_ m$, and with the sets $O_{3m}$ (shrunk to $O_{3m+1}$), $O_{3m+2}$, $O_{3m+3}$ in the roles of $O_1$, $O$, $O'$; this bounds $\| u\| _{W^{m+1,p_{m+1}}(O_{3m+3})}$ in terms of $\| u\| _{W^{m,p_ m}(O_{3m})}$. For $m=\ell -1$ we obtain the bound for $\| u\| _{W^{\ell ,p}(V)}$. $\square$

Proof. By the inversion formula of theorem 2.9.4 and Fubini's theorem, $\widehat{vw}=\widehat v*\widehat w$. With $\langle \xi \rangle =(1+|\xi |^2)^{1/2}$ we have $\langle \xi \rangle \leq \langle \eta \rangle +\langle \xi -\eta \rangle $, since $(1,\xi )=(1,\eta )+(0,\xi -\eta )$ in $\mathbb R^{1+d}$; hence $\langle \xi \rangle ^ s\leq 2^ s(\langle \eta \rangle ^ s+\langle \xi -\eta \rangle ^ s)$, and therefore

Young's inequality $\| F*G\| _2\leq \| F\| _2\| G\| _1$ follows from the Cauchy–Schwarz inequality $|F*G(x)|^2\leq \| G\| _1\int |F(x-y)|^2|G(y)|\, \mathrm dy$ by integration in $x$ and Fubini's theorem. It therefore remains to bound $\| \widehat w\| _1$ and $\| \widehat v\| _1$ by the norms $\| \cdot \| _{\langle s\rangle }$. This follows from the Cauchy–Schwarz inequality, $\| \widehat w\| _1\leq \| \langle \cdot \rangle ^{-s}\| _2\| \langle \cdot \rangle ^ s\widehat w\| _2$, since $\langle \xi \rangle ^{-2s}$ is integrable on $\mathbb R^ d$ for $2s>d$. $\square$

Proof. For $0\leq t\leq 1$ consider the equation

and let $T\subset [0,1]$ be the set of $t$ for which it has a smooth real solution. For $t=0$ the solution is $a=0$, so $T\neq \varnothing $. We show that $T$ is open and closed in $[0,1]$; then $T=[0,1]$, since $[0,1]$ is connected, and the case $t=1$ gives existence. Uniqueness is proved at the end.

Step 1: a priori estimates. Let $a$ be a smooth solution for some $t\in [0,1]$, with metric $g$. In this step, constants depend only on $X$, $\omega $, $\kappa $, $\beta $ and $f$, but not on $t$ or on the solution. Since $\| tf\| _\infty \leq \| f\| _\infty $ and, by (16.15.1), $\sup _ XD_ h(tf)=t\sup _ XD_ hf\leq \sup _ XD_ hf$, lemma 16.15.7, applied to $tf$, gives $|a|\leq \| f\| _\infty $ and $C_0^{-1}h\leq g\leq C_0h$ with $C_0$ independent of $t$. Hence $La=-4\pi D_ ha=-4\pi (H-\operatorname {tr}_ hk)$ is bounded, and proposition 16.14.9 bounds $a$ in $W^{2,p}(X)$ for every finite $p$; by (16.14.3) and Morrey's inequality, this bounds $\| a\| _{C^1}$ and $\| D^2a\| _{L^ p}$ on compact subsets of charts. By lemma 16.15.13, the first derivatives of the $g_{i\bar j}$ are bounded on compact subsets of the charts of a fixed finite atlas.

Fix finitely many coordinate balls $U_\nu $ whose closures lie in chart domains, and concentric balls $V'_\nu \Subset V_\nu \Subset U_\nu $ such that the $V'_\nu $ cover $X$. In the coordinates of $U_\nu $, the function $a$ solves (16.15.18) with $\phi _0=\log \det k-\beta tf$, whose derivatives are bounded independently of $t$. By the preceding paragraph, on $U_\nu $ the matrix $G=g$ lies between two fixed positive multiples of the identity, it is Lipschitz with a fixed constant, so that we may take $\varpi (r)=Cr$, and $\| a\| _{C^1(U_\nu )}+\| D^2a\| _{L^ p(U_\nu )}$ is bounded for every finite $p$. By lemma 16.15.17, $\| a\| _{W^{\ell ,p}(V_\nu )}$ is bounded for all $\ell \geq 3$ and $p<\infty $, and by Morrey's inequality every derivative of $a$ is bounded on $V'_\nu $. Thus every $C^\ell $ norm of $a$, computed in these coordinates on the sets $V'_\nu $, is bounded independently of $t$.

Step 2: openness. Let $t_0\in T$, with smooth solution $a$ and metric $g$; thus $\kappa +\mathrm{dd}^{\mathrm{c}}a$ is a Kähler form. For $t$ near $t_0$ we look for a solution of the form $a+b$, and we find $b$ by the contraction mapping principle in a Sobolev space. Fix an integer $s>n$. We use the spaces $H^ r$ of section 10.2, defined with the coordinate balls $U_\nu $ of Step 1 and a partition of unity $(\chi _\nu )$ subordinate to them; by lemma 10.2.1(1), other choices give equivalent norms. For a smooth function $v$ with compact support in $U_\nu $, written in the coordinates of $U_\nu $, let $\| v\| _{\langle r\rangle }$ be defined by the formula of lemma 16.15.19 with $d=2n$, so that $\| u\| _{H^ r}^2=\sum _\nu \| \chi _\nu u\| _{\langle r\rangle }^2$ for smooth $u$. Choose smooth functions $\tilde\chi _\nu $, $\check\chi _\nu $ on $X$ with values in $[0,1]$ and compact supports in $U_\nu $ such that $\tilde\chi _\nu =1$ near $\operatorname{Supp}\chi _\nu $ and $\check\chi _\nu =1$ near $\operatorname{Supp}\tilde\chi _\nu $. We use three facts.

1. Since $s>n$, lemma 10.2.1(3) gives continuous inclusions $H^{s+2}\subset C^2(X)$ and $H^ s\subset C^0(X)$.

2. For a smooth function $\psi $ with compact support in $U_\nu $, multiplication by $\psi $ is bounded for $\| \cdot \| _{\langle s\rangle }$, by lemma 16.15.19; and $\| \partial _\mu v\| _{\langle r\rangle }\leq 2\pi \| v\| _{\langle r+1\rangle }$ for real coordinate derivatives, since $\widehat{\partial _\mu v}=2\pi \mathrm i\xi _\mu \widehat v$.

3. $\| \check\chi _\nu b\| _{\langle s+2\rangle }\leq C\| b\| _{H^{s+2}}$ for smooth $b$. Indeed, $\Sigma =\sum _\iota \check\chi _\iota \geq 1$, because $\check\chi _\iota =1$ wherever $\chi _\iota >0$; so the functions $\check\chi _\iota /\Sigma $ form a partition of unity subordinate to the same charts, and by lemma 10.2.1(1) the norm of $H^{s+2}$ computed with them is equivalent to the given one. Moreover $\check\chi _\nu b=\eta _\nu \cdot (\check\chi _\nu /\Sigma )b$, where $\eta _\nu $ is a smooth function with compact support in $U_\nu $ equal to $\Sigma $ near $\operatorname{Supp}\check\chi _\nu $, and multiplication by $\eta _\nu $ is bounded for $\| \cdot \| _{\langle s+2\rangle }$ by lemma 16.15.19.

For a real function $b\in C^2(X)$, let $Y$ be the tensor $Y^ m{}_ j=g^{m\bar l}b_{j\bar l}$; its powers $(Y^ q)^ m{}_ j=Y^ m{}_{a_1}Y^{a_1}{}_{a_2}\cdots Y^{a_{q-1}}{}_ j$ are tensors, so the traces $\operatorname {tr}(Y^ q)$ are functions on $X$. In a chart, with the matrices $G=(g_{i\bar j})$ and $B=(b_{i\bar j})$, the traces of the powers of $Y$ are those of $BG^{-1}$, which is similar to the Hermitian matrix $G^{-1/2}BG^{-1/2}$; let $\lambda _1,\ldots ,\lambda _ n$ be their real eigenvalues. Then $\det (G+B)=\det G\prod _ i(1+\lambda _ i)$ and $\operatorname {tr}Y=\sum _ i\lambda _ i=g^{i\bar j}b_{i\bar j}=D_ gb$. Since $|\lambda _ i|\leq \| G^{-1}\| \, |B|$, and $G$ is bounded below on compact subsets of the charts of a finite atlas, (i) gives $r_0>0$ such that $\| b\| _{H^{s+2}}\leq r_0$ implies $|\lambda _ i|\leq 1/2$; then $G+B>0$, that is, $\kappa +\mathrm{dd}^{\mathrm{c}}(a+b)>0$, and

Put $\mathcal N_ t(a+b)=\log (\det (G+B)/\det k)-\beta (a+b-tf)$, so that $a+b$ solves the equation for $t$ exactly when $\mathcal N_ t(a+b)=0$. Since $\log (\det G/\det k)=\beta (a-t_0f)$,

We estimate $\mathcal Q$ in $H^ s$. Let $b$ be smooth and real with $\| b\| _{H^{s+2}}\leq r_0$. In the chart $U_\nu $ let $Y_\nu =\tilde\chi _\nu Y$, a matrix of smooth functions with compact support, and $\| Y_\nu \| _{\langle s\rangle }=\max _{m,j}\| \tilde\chi _\nu Y^ m{}_ j\| _{\langle s\rangle }$. Since $\tilde\chi _\nu Y^ m{}_ j=(\tilde\chi _\nu g^{m\bar l})\, \partial _ j\partial _{\bar l}(\check\chi _\nu b)$, facts (ii) and (iii) give $\| Y_\nu \| _{\langle s\rangle }\leq C_ Y\| b\| _{H^{s+2}}$. For matrices $M$, $N$ of smooth compactly supported functions, lemma 16.15.19 gives $\| (MN)_{mj}\| _{\langle s\rangle }\leq nC_ s\max \| M_{\cdot \cdot }\| _{\langle s\rangle }\max \| N_{\cdot \cdot }\| _{\langle s\rangle }$, and therefore

where $C_\chi $ bounds multiplication by $\chi _\nu $. Since $Y_\nu =Y$ on $\operatorname{Supp}\chi _\nu $, the function $\chi _\nu \mathcal Q(b)$ is the sum of the series $\sum _{q\geq 2}\frac{(-1)^{q+1}}q\chi _\nu \operatorname {tr}(Y_\nu ^ q)$, which converges uniformly, with supports in $\operatorname{Supp}\chi _\nu $. Decreasing $r_0$, we may assume that $nC_ sC_ Yr_0\leq 1/2$. Then the series converges absolutely for $\| \cdot \| _{\langle s\rangle }$; since its partial sums converge in $L^2$, their Fourier transforms converge in $L^2$ (theorem 2.9.4), a subsequence converges almost everywhere, and Fatou's lemma gives

Summing over $\nu $, we obtain $\| \mathcal Q(b)\| _{H^ s}\leq C_ Q\| b\| _{H^{s+2}}^2$. In the same way, using $Y^ q-Y'^ q=\sum _{i=0}^{q-1}Y^ i(Y-Y')Y'^{q-1-i}$, we obtain

for two such functions $b$, $b'$.

Both estimates extend to real elements of $H^{s+2}$. Let $H^{s+2}_{\mathbb R}$ be the closed subspace of real elements of $H^{s+2}$. Smooth real functions are dense in it, since taking real parts does not increase the norms: $|\widehat{\chi \operatorname{Re}v}(\xi )|^2\leq \frac12(|\widehat{\chi v}(\xi )|^2+|\widehat{\chi v}(-\xi )|^2)$ for real $\chi $. If $b\in H^{s+2}_{\mathbb R}$ with $\| b\| _{H^{s+2}}<r_0$, and smooth real $b_ j$ converge to $b$ in $H^{s+2}$, with norms less than $r_0$, then the $\mathcal Q(b_ j)$ form a Cauchy sequence in $H^ s$; by (i), $b_ j\to b$ in $C^2$, so $\mathcal Q(b_ j)\to \mathcal Q(b)$ uniformly, and, again by (i), the limit in $H^ s$ is $\mathcal Q(b)$. Hence the two estimates for $\mathcal Q$ hold on the ball $\| b\| _{H^{s+2}}<r_0$ of $H^{s+2}_{\mathbb R}$.

The operator $\mathcal D\colon H^{s+2}\to H^ s$ is invertible, with bounded inverse. Indeed, $\kappa +\mathrm{dd}^{\mathrm{c}}a=\frac{\mathrm i}{2\pi }g_{i\bar j}\, \mathrm dz_ i\wedge \mathrm d\bar z_ j$ is a Kähler form, and by (10.2.14), applied with $\omega '=\kappa +\mathrm{dd}^{\mathrm{c}}a$, we have $D_ g-\beta =-(4\pi )^{-1}(L_{\omega '}+4\pi \beta )$. Since $\omega $ is an arbitrary Kähler form on $X$, lemma 10.2.5(3) holds for $\omega '$; applied to functions, on which the Hodge Laplacian of $\omega '$ is $L_{\omega '}$, with $c=4\pi \beta >0$, it shows that $D_ g-\beta $ maps $H^{s+2}$ isomorphically onto $H^ s$, because these spaces do not depend on the Kähler form (lemma 10.2.1(4)). Let $M_{\mathcal D}$ be the norm of its inverse. Since $\mathcal D$ is a real operator, its inverse maps real elements to real elements. For $b\in H^{s+2}$, the element $\mathcal Db$ of $H^ s$ is the continuous function $D_ gb-\beta b$, by approximation with smooth functions and (i).

The solutions $b\in H^{s+2}_{\mathbb R}$ of $\mathcal N_ t(a+b)=0$ with $\| b\| _{H^{s+2}}<r_0$ are the fixed points of

Choose $0<r<r_0$ with $4M_{\mathcal D}C_ Qr\leq 1$, and suppose that $|t-t_0|\leq \delta $, where $2M_{\mathcal D}\beta \delta \| f\| _{H^ s}\leq r$. On the closed ball $\overline B_ r$ of radius $r$ in $H^{s+2}_{\mathbb R}$ we then have $\| \Phi _ t(b)\| _{H^{s+2}}\leq M_{\mathcal D}(\beta \delta \| f\| _{H^ s}+C_ Qr^2)\leq r/2+r/4$ and $\| \Phi _ t(b)-\Phi _ t(b')\| _{H^{s+2}}\leq 2M_{\mathcal D}C_ Qr\| b-b'\| _{H^{s+2}}\leq \frac12\| b-b'\| _{H^{s+2}}$. The iterates $b^{(0)}=0$, $b^{(j+1)}=\Phi _ t(b^{(j)})$ therefore stay in $\overline B_ r$, their successive differences decrease geometrically, so they form a Cauchy sequence, and their limit $b$ is a fixed point of $\Phi _ t$, by the Lipschitz bound. Thus $\mathcal Db=-\beta (t-t_0)f-\mathcal Q(b)$ in $H^ s$, hence pointwise by (i), that is, $\mathcal N_ t(a+b)=0$. The function $u=a+b$ is of class $C^2$, satisfies $\kappa +\mathrm{dd}^{\mathrm{c}}u>0$, and in every chart it solves (16.15.18) with $\phi _0=\log \det k-\beta tf$. By lemma 16.15.17, $u$ is smooth. Hence $T$ contains $[t_0-\delta ,t_0+\delta ]\cap [0,1]$, and $T$ is open in $[0,1]$.

Step 3: closedness. Let $t_ j\in T$ converge to $t_*\in [0,1]$, with solutions $a_ j$. By Step 1, every $C^\ell $ norm of $a_ j$ on the compact sets $\overline{V'_\nu }$ is bounded independently of $j$. Hence, for every $\ell $, the derivatives of order at most $\ell $ of the $a_ j$ are uniformly bounded on each $\overline{V'_\nu }$, and they are equicontinuous there by the mean value theorem, since the derivatives of order $\ell +1$ are bounded and the balls $V'_\nu $ are convex. By the Arzelà–Ascoli theorem (proposition 2.7.4) and a diagonal argument over $\ell $ and $\nu $, a subsequence converges, together with all its derivatives, uniformly on each $\overline{V'_\nu }$. The limit $a_*$ is smooth, and its derivatives are the limits of the derivatives of the $a_ j$, because uniform convergence of functions and of their first derivatives on a ball implies that the limit is continuously differentiable with the limit derivatives. The uniform lower bound $g_ j\geq C_0^{-1}h$ of Step 1 gives $\kappa +\mathrm{dd}^{\mathrm{c}}a_*>0$, and $a_*$ solves the equation for $t_*$. Thus $t_*\in T$, and $T$ is closed.

Step 4: uniqueness. Let $a$ and $a'$ be smooth solutions of (16.15.8). At a maximum point of $a-a'$ the complex Hessian of $a-a'$ is negative semidefinite, so $0<k+(a_{i\bar j})\leq k+(a'_{i\bar j})$, and the equations give $\mathrm e^{\beta (a-f)}\leq \mathrm e^{\beta (a'-f)}$ there. Hence $a-a'\leq 0$ at that point, and therefore everywhere. Interchanging $a$ and $a'$ proves that $a=a'$. $\square$

Proof. At each point, the eigenvalues of $k$ relative to $h$ are positive, and the arithmetic–geometric mean inequality gives $\operatorname {tr}_ hk\geq n(\det k/\det h)^{1/n}\geq nd_0^{1/n}$. Since $\sup _ XD_ hf\geq 0$ by (16.15.1), it follows that $B\geq nd_0^{1/n}$, that is, $K_0\geq 1$.

The upper bound. Let $\beta >0$, write $a=a_\beta $ and $g=k+(a_{i\bar j})$, and let $x_0$ be a maximum point of $a-f$. There the complex Hessian of $a-f$ is negative semidefinite, so $0<g\leq k+(f_{i\bar j})$. In particular $k+(f_{i\bar j})$ is positive definite at $x_0$, and the arithmetic–geometric mean inequality for its eigenvalues relative to $h$ gives

at $x_0$. Hence $a-f\leq \log K_0/\beta $ on $X$. The function $a-\log K_0/\beta $ is therefore a smooth $\kappa $-psh function below $f$, that is, a competitor for $P_\kappa (f)$, and so $a\leq P_\kappa (f)+\log K_0/\beta $.

The lower bound. Let $\beta \geq 1$ and put $v=P_\kappa (f)$; as noted at the beginning of this subsection, $\min _ Xf\leq v\leq f$. For $0<t\leq 1$ put

This is a bounded $\kappa $-psh function, and we show that $w\leq a$.

For a bounded $\kappa $-psh function $b$ write $\mu _ b=\langle (\kappa +\mathrm{dd}^{\mathrm{c}}b)^ n\rangle $, the non-pluripolar Monge–Ampère measure, which is the Bedford–Taylor measure since $b$ is bounded (proposition 13.4.3, applied with $q=p=n$ to the sums of $b$ with smooth local potentials of $\kappa $), and equals the smooth form $(\kappa +\mathrm{dd}^{\mathrm{c}}b)^ n$ if $b$ is smooth (proposition 7.2.2(1)). Locally $\kappa =\mathrm{dd}^{\mathrm{c}}\rho $ with $\rho $ smooth, and $\kappa +\mathrm{dd}^{\mathrm{c}}w=\mathrm{dd}^{\mathrm{c}}\bigl ((1-t)(\rho +v)+t\rho \bigr )$. Expanding the $n$th power by multilinearity and positive homogeneity (proposition 13.2.6), and discarding the mixed terms, which are positive measures, we get $\mu _ w\geq t^ n\kappa ^ n$. On the other hand $w\leq f-n\log (1/t)/\beta $, so $\mathrm e^{\beta (w-f)}\leq t^ n$.

Let $E=\{ a<w\} $, a Borel set. Since $a$ and $w$ are bounded, $a\geq w-O(1)$, so proposition 13.9.5 applies without the correction terms; together with the two inequalities just proved and the equation for $a$ it gives

On $E$ the first integrand is strictly larger than the last one, so $E$ has volume zero. Thus $w\leq a$ almost everywhere, and hence everywhere by lemma 6.14.4(1), since both functions are quasi-plurisubharmonic.

Now take $t=1/\beta $, which is allowed since $\beta \geq 1$. Since $\min _ Xf-v\geq \min _ Xf-\max _ Xf=-\operatorname {osc}_ Xf$, we obtain

The Laplacian bound. Let $\beta _0$ be the constant of lemma 16.15.7. For $\beta >\beta _0$ that lemma, together with $D_ ha_\beta =H-\operatorname {tr}_ hk$, $H>0$ and $\operatorname {osc}_ Xf\leq 2\| f\| _\infty $, gives

Since $a_\beta \to v$ uniformly, $\langle D_ hv,\chi \rangle =\int _ Xv\, D_ h\chi \, \mathrm dV=\lim _{\beta \to \infty }\int _ X(D_ ha_\beta )\chi \, \mathrm dV$ for smooth $\chi $, which proves (16.15.22). By the remark following (16.15.2), $Lv=-4\pi D_ hv$ is therefore a bounded function, with a bound depending only on $X$, $\omega $, $\kappa $ and upper bounds for $\| f\| _\infty $ and $\sup _ XD_ hf$. Finally $v$ is continuous, as a uniform limit of continuous functions, so by proposition 16.14.9 it belongs to $C^{1,\alpha }(X)$ for every $0<\alpha <1$. $\square$

Proof. Put

The weights $p_ j$ are positive with $\sum _ jp_ j=1$, and $\min _ jf_ j-\delta \log r\leq F\leq \min _ jf_ j$, because the sum lies between $\mathrm e^{-\min _ jf_ j/\delta }$ and $r$ times this number. In local holomorphic coordinates, the chain rule gives $\partial _ iF=\sum _ jp_ j\partial _ if_ j$ and $\partial _{\bar m}p_ j=-\delta ^{-1}p_ j\bigl (\partial _{\bar m}f_ j-\sum _ lp_ l\partial _{\bar m}f_ l\bigr )$; differentiating once more and contracting with $h^{i\bar m}$, we get

The term in parentheses equals $\sum _ jp_ j|\partial f_ j-\sum _ lp_ l\partial f_ l|_ h^2\geq 0$, the variance of the vectors $\partial f_ j$ with respect to the probability weights $p_ j$. Hence $D_ hF\leq \sum _ jp_ jD_ hf_ j\leq \max _ j\sup _ XD_ hf_ j$. $\square$

Proof. We approximate $\min _ jf_ j$ uniformly by smooth obstacles to which (16.15.22) applies with uniform constants, and pass to the limit.

Step 1: smoothing the obstacles. By proposition 16.14.9, each $f_ j$ belongs to $W^{2,p}(X)$ for every finite $p$, and it is of class $C^1$; fix $p>2n$. Choose a finite smooth partition of unity $(\chi _\nu )$ with each $\chi _\nu $ supported in the interior of a coordinate chart. For small $\varepsilon >0$, let $f_{j,\nu ,\varepsilon }$ be the convolution of $f_ j$, written in the chart of $\chi _\nu $, with the mollifier $\rho _\varepsilon $ of section 2.5; it is defined on a neighborhood of the support of $\chi _\nu $. Put $f_{j,\varepsilon }=\sum _\nu \chi _\nu f_{j,\nu ,\varepsilon }$. These are smooth functions with $\| f_{j,\varepsilon }\| _\infty \leq \| f_ j\| _\infty $, and $f_{j,\varepsilon }\to f_ j$ in $C^1$, because the convolutions of the $C^1$ function $f_ j$ and of its first derivatives converge to them uniformly on compact sets (lemma 2.5.1(1)), and $\partial _\mu (f_ j*\rho _\varepsilon )=(\partial _\mu f_ j)*\rho _\varepsilon $.

We bound the Laplacians of the approximations. By (16.14.8), in the real coordinates $x_1,\ldots ,x_{2n}$ underlying a holomorphic chart, with $\partial _\mu =\partial /\partial x_\mu $, we have $D_ h=\sum _{\mu ,\sigma }c^{\mu \sigma }(x)\partial _\mu \partial _\sigma $ on real functions, with smooth real coefficients; there are no first-order terms. Since $f_ j\in W^{2,p}_{\mathrm{loc}}$ in the chart, by (16.14.3), and $\partial _\mu \partial _\sigma (f_ j*\rho _\varepsilon )=(\partial _\mu \partial _\sigma f_ j)*\rho _\varepsilon $, the commutator of $D_ h$ with convolution is

where $D_ hf_ j=-Lf_ j/(4\pi )$ is a bounded function. Since $|c^{\mu \sigma }(x)-c^{\mu \sigma }(y)|\leq C\varepsilon $ on the support of $\rho _\varepsilon (x-\cdot )$ and $\| \rho _\varepsilon \| _{L^{p'}}=\varepsilon ^{-2n/p}\| \rho \| _{L^{p'}}$ with $p'=p/(p-1)$, Hölder's inequality and (16.14.3) bound this commutator by $C\varepsilon ^{1-2n/p}\| f_ j\| _{W^{2,p}(X)}$, which tends to zero because $p>2n$. As $(D_ hf_ j)*\rho _\varepsilon $ lies between the essential infimum and the essential supremum of $D_ hf_ j$, since $\rho _\varepsilon \geq 0$ has integral one, we obtain on a neighborhood of the support of $\chi _\nu $

Next, $D_ hf_{j,\varepsilon }=\sum _\nu \chi _\nu D_ hf_{j,\nu ,\varepsilon }+\sum _\nu [D_ h,\chi _\nu ]f_{j,\nu ,\varepsilon }$, where each commutator $[D_ h,\chi _\nu ]$ is a first-order operator with coefficients supported in $\operatorname{Supp}\chi _\nu $. Since $\sum _\nu \chi _\nu =1$, we have $\sum _\nu [D_ h,\chi _\nu ]f_ j=D_ hf_ j-D_ hf_ j=0$, so the second sum equals $\sum _\nu [D_ h,\chi _\nu ](f_{j,\nu ,\varepsilon }-f_ j)$, which involves only $f_{j,\nu ,\varepsilon }-f_ j$ and its first derivatives and tends to zero uniformly. The first sum is a convex combination. Hence $|D_ hf_{j,\varepsilon }|\leq \| D_ hf_ j\| _{L^\infty }+1$ for all sufficiently small $\varepsilon $.

Step 2: smoothing the minimum. For $\delta >0$ set

By lemma 16.15.23,

Choose sequences $\varepsilon _ m,\delta _ m\to 0$ with $\varepsilon _ m$ small as in Step 1, and put $F_ m=F_{\varepsilon _ m,\delta _ m}$. Then $F_ m\to \min _ jf_ j$ uniformly on $X$, since $|\min _ jf_{j,\varepsilon _ m}-\min _ jf_ j|\leq \max _ j\| f_{j,\varepsilon _ m}-f_ j\| _\infty $, while $\| F_ m\| _\infty \leq \max _ j\| f_ j\| _\infty +\delta _ m\log r$ and $\sup _ XD_ hF_ m\leq \max _ j\| D_ hf_ j\| _{L^\infty }+1$ remain bounded.

Step 3: passage to the limit. By (16.15.22), applied to the smooth obstacles $F_ m$, we have $-\sup _ X\operatorname {tr}_ hk\leq D_ hP_\kappa (F_ m)\leq 2B_ m\mathrm e^{2A\| F_ m\| _\infty }$ in the sense of distributions, where $A$ depends only on $X$, $\omega $, $\kappa $, and $B_ m=\sup _ X\operatorname {tr}_ hk+\sup _ XD_ hF_ m$. By Step 2 these bounds are uniform in $m$ and depend only on the asserted data. By the stability of envelopes in proposition 16.11.4(4), $\| P_\kappa (F_ m)-u\| _\infty \leq \| F_ m-\min _ jf_ j\| _\infty \to 0$. Therefore $D_ hP_\kappa (F_ m)\to D_ hu$ in the sense of distributions, and $u$ satisfies (16.15.2) with $c_1=\sup _ X\operatorname {tr}_ hk$ and $c_2=\sup _ m2B_ m\mathrm e^{2A\| F_ m\| _\infty }$. By the remark following (16.15.2), $D_ hu$, and hence $Lu=-4\pi D_ hu$, is a bounded function with the asserted bound.

The function $u$ is continuous, as a uniform limit of the functions $P_\kappa (F_ m)$, which are continuous by proposition 16.15.21; so proposition 16.14.9 shows that $u\in C^{1,\alpha }(X)$ for every $0<\alpha <1$. Finally, $u$ is a continuous $\kappa $-psh function with bounded metric Laplacian, so by proposition 16.14.6, applied with $U=X$, $\theta _ j=\kappa $, and $u$ in place of both $u_ j$ and $f_ j$ for every $j$, the coefficients of $\kappa +\mathrm{dd}^{\mathrm{c}}u$ are bounded on every coordinate ball $W\Subset X$, hence on $X$. $\square$