Uniform $C^{1,\alpha }$ bounds control averages of the trace $\operatorname {tr}_\beta (\vartheta +\mathrm{dd}^{\mathrm{c}}u_ j)=\operatorname {tr}_\beta \vartheta +\Delta _\beta u_ j$ over small balls. To pass from averages to pointwise values we need a bound for the gradient of the trace, that is, a third-order estimate, whose dependence on the ellipticity constants is polynomial. We prove it by Calabi's maximum-principle argument, computed with the Chern connection of the solution so that the torsion of the nonclosed background appears explicitly. The resulting averaging inequality is nonlinear in the trace, and it closes under iteration when $\alpha $ is sufficiently close to one.
In this subsection $B$ is the unit ball and $B_ r(p)$ is the Euclidean ball of radius $r$ centered at $p$. For a Hermitian metric $g$, $\partial g$ denotes the ordinary coordinate derivatives of its coefficient matrix, while tensor derivatives in the proofs are taken with respect to the Chern connection of $g$.
Proof.
The quantity $\mathcal R$ below is the squared norm of the difference between the Chern connection of $g$ and the flat connection. We derive a differential inequality for $\mathcal R$ in which the torsion enters through fixed derivatives of $\vartheta $, and combine it with the trace in a maximum principle; every constant is kept a power of $M$.
We compare the flat connection of $\beta $ with the Chern connection $\nabla $ of $g$. Write $g^{k\bar\ell }$ for the coefficients of the inverse matrix and $\bar\nabla $ for covariant differentiation in $(0,1)$ directions; tensor norms and inner products without subscript are computed with $g$. The difference of the two connections and its squared norm are
\[ \Psi ^ k_{ij}=g^{k\bar\ell }\partial _ i g_{j\bar\ell }, \qquad \mathcal R=|\Psi |_ g^2. \]
The torsion of $\nabla $ is determined by the fixed background, because the third derivatives of $u$ cancel in it. Its lowered and raised components are
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\begin{equation} C_{ij\bar\ell } =\partial _ i\vartheta _{j\bar\ell }-\partial _ j\vartheta _{i\bar\ell }, \qquad \mathcal T^ k_{ij}=g^{k\bar\ell }C_{ij\bar\ell }. \end{equation}
Consequently, for a fixed integer $N$,
14.39
\begin{align} |\mathcal T|& \leq CM^ N,\nonumber \\ |\nabla \mathcal T|+|\bar\nabla \mathcal T| & \leq CM^ N(1+\sqrt{\mathcal R}),\tag{15.18.39}\\ |\nabla ^2\mathcal T|+|\nabla \bar\nabla \mathcal T|+|\bar\nabla ^2\mathcal T| & \leq CM^ N(1+\mathcal R+|\nabla \Psi |+|\bar\nabla \Psi |).\nonumber \end{align}
Every factor here is either a fixed derivative of $\vartheta $, a connection factor $\Psi $, or a factor of $g^{\pm 1}$; this explains the polynomial dependence.
For tensor fields, use $\Delta _ g=g^{p\bar q}\nabla _ p\nabla _{\bar q}$. We need the following curvature identities. With
\[ \operatorname {Rm}^ k{}_{i\bar j l}=-\partial _{\bar j}\Psi ^ k_{il}, \qquad \operatorname {Rm}_{i\bar j l\bar k}=g_{m\bar k}\operatorname {Rm}^ m{}_{i\bar j l}, \]
set
\[ \operatorname {Ric}^{(2)}_{l\bar k}=g^{i\bar j}\operatorname {Rm}_{i\bar j l\bar k}, \qquad \operatorname {Ric}^{(1)}_{l\bar k}=g^{i\bar j}\operatorname {Rm}_{l\bar k i\bar j} =-\partial _ l\partial _{\bar k}\log F. \]
The two curvature interchanges are
\begin{align*} \operatorname {Rm}_{i\bar j l\bar k}-\operatorname {Rm}_{l\bar j i\bar k} & =-\nabla _{\bar j}C_{il\bar k},\\ \operatorname {Rm}_{i\bar j l\bar k}-\operatorname {Rm}_{i\bar k l\bar j} & =-\nabla _ i\overline{C}_{\bar j\bar k l}. \end{align*}
They follow by differentiating the torsion identity and its conjugate. Successive interchange and contraction give
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\begin{equation} \operatorname {Ric}^{(2)}_{l\bar k} =\operatorname {Ric}^{(1)}_{l\bar k} -g^{i\bar j}\bigl ( \nabla _ l\overline C_{\bar j\bar k i} +\nabla _{\bar j}C_{il\bar k} \bigr ). \end{equation}
Metric compatibility, (15.18.39), and the fixed smooth formula for $\operatorname {Ric}^{(1)}$ imply
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\begin{equation} |\operatorname {Ric}^{(2)}|\leq CM^ N(1+\sqrt{\mathcal R}), \qquad |\nabla \operatorname {Ric}^{(2)}| \leq CM^ N(1+\mathcal R+|\nabla \Psi |+|\bar\nabla \Psi |). \end{equation}
Moreover,
\[ \nabla _{\bar q}\Psi ^ k_{ij}=-\operatorname {Rm}^ k{}_{i\bar q j}, \]
and the Chern–Bianchi identity, including its torsion term, is
\[ \nabla _ p\operatorname {Rm}^ k{}_{i\bar q j}-\nabla _ i\operatorname {Rm}^ k{}_{p\bar q j} =-\mathcal T^ r_{pi}\operatorname {Rm}^ k{}_{r\bar q j}. \]
Consequently,
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\begin{equation} \Delta _ g\Psi ^ k_{ij} =-\nabla _ i(\operatorname {Ric}^{(2)})^ k{}_ j +g^{p\bar q}\mathcal T^ r_{pi}\operatorname {Rm}^ k{}_{r\bar q j}. \end{equation}
If $A^ k{}_ l=(\operatorname {Ric}^{(2)})^ k{}_ l$, define its action on a tensor of type $(1,2)$ by
\[ (A\mathbin {\cdot }\Psi )^ k_{ij} =A^ k{}_ l\Psi ^ l_{ij}-A^ l{}_ i\Psi ^ k_{lj}-A^ l{}_ j\Psi ^ k_{il}. \]
Differentiating the squared norm and using metric compatibility gives
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\begin{equation} \Delta _ g\mathcal R =|\nabla \Psi |^2+|\bar\nabla \Psi |^2 +2\operatorname {Re}\langle \Delta _ g\Psi ,\Psi \rangle -\operatorname {Re}\langle A\mathbin {\cdot }\Psi ,\Psi \rangle . \end{equation}
There is no mixed torsion term because the Chern torsion has type $(2,0)$. Using (15.18.39), (15.18.41), and (15.18.42) in (15.18.43), Young's inequality absorbs the products of $\sqrt{\mathcal R}$ with $|\nabla \Psi |+|\bar\nabla \Psi |$. We obtain
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\begin{equation} \Delta _ g\mathcal R\geq -CM^ N(\mathcal R^{3/2}+1). \end{equation}
Let $s=\operatorname {tr}_\beta g$. Direct differentiation as in (15.18.10) gives
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\begin{equation} \Delta _ gs\geq M^{-1}\mathcal R-CM, \qquad |\partial s|_ g^2\leq nM^2\mathcal R. \end{equation}
Take a cutoff $\zeta $ supported in $B$, equal to one on $B_{1/2}$, set $\eta =\zeta ^2$ and $h=2nM-s$, so $nM\leq h\leq 2nM$, and maximize
\[ Q=\frac{\eta \mathcal R}{h}. \]
At a positive interior maximum of $Q$, the identity $\partial \mathcal R/\mathcal R=\partial h/h-\partial \eta /\eta $ gives
\[ \begin{split} 0\geq \frac{\Delta _ gQ}{Q} & =\frac{\Delta _ g\mathcal R}{\mathcal R} +\frac{\Delta _ g\eta }{\eta }-\frac{\Delta _ gh}{h} -\frac{2|\partial \eta |_ g^2}{\eta ^2} +\frac{2\operatorname {Re}\langle \partial \eta ,\partial h\rangle _ g}{\eta h}. \end{split} \]
We have $|\partial \eta |_ g^2\leq CM\eta $, $|\Delta _ g\eta |\leq CM$ and $|\partial h|_ g\leq \sqrt n M\sqrt{\mathcal R}$. Insert (15.18.44) and (15.18.45) and multiply by $\eta Q$. The positive term is $M^{-1}Q^2$. The two terms involving $\sqrt{\mathcal R}$ are bounded in absolute value by a fixed power of $M$ times $Q^{3/2}$, since $\mathcal R=hQ/\eta $ and $h\leq 2nM$. The remaining cutoff terms are bounded by $CM^ N(Q+1)$. Therefore
\[ 0\geq M^{-1}Q^2-CM^ N(Q^{3/2}+Q+1). \]
Thus $Q$, and then $\mathcal R$ on $B_{1/2}$, are bounded by a fixed polynomial in $M$. Since $|\partial g|_\beta ^2\leq M^3\mathcal R$, this proves (15.18.36).
If $s\leq L$, the determinant lower bound gives $\lambda _{\min }(g)\geq (\inf F)L^{1-n}$, so $M$ is a fixed power of $1+L$. For a ball $B_ r(p)$, set $u_ r(z)=r^{-2}u(p+rz)$ and use the coefficient matrices $\vartheta _ r(z)=\vartheta (p+rz)$, $g_ r(z)=g(p+rz)$ and $F_ r(z)=F(p+rz)$. Then $g_ r=\vartheta _ r+\mathrm{dd}^{\mathrm{c}}u_ r$ in the unit coordinates and $\det g_ r=F_ r$. For $r\leq 1$ the derivatives of the fixed data remain uniformly bounded. The unit-ball estimate for the ordinary coefficient derivatives $\partial g_ r$ and their conjugates gives the factor $r^{-1}$ in (15.18.37).
$\square$
Proof.
Fix a center for concentric balls $B_ r\Subset B_0$, put $s_ j=\operatorname {tr}_\beta (\vartheta +\mathrm{dd}^{\mathrm{c}}u_ j)$, and set
\[ M_ j(r)=1+\sup _{B_ r}s_ j. \]
For concentric balls $B_ r\Subset B_ R\Subset B_0$, $d=R-r\leq 1$, proposition 15.18.35 gives
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\begin{equation} |\nabla s_ j|\leq Cd^{-1}M_ j(R)^ N \quad \text{on }B_{r+d/2}. \end{equation}
Fix $p\in B_ r$ and let $\chi _ h$ be a nonnegative mollifier of radius $h\leq d/4$. Since
\[ s_ j=\operatorname {tr}_\beta \vartheta +\Delta _\beta u_ j, \]
integration by parts, after subtracting the constant vector $\nabla u_ j(p)$, and the uniform $C^{1,\alpha }$ bound give
\[ (s_ j*\chi _ h)(p)\leq C(1+h^{\alpha -1}). \]
Equation (15.18.47) gives
\[ |s_ j(p)-(s_ j*\chi _ h)(p)| \leq Cd^{-1}M_ j(R)^ Nh. \]
Choose
\[ h=\frac d4M_ j(R)^{-N/(2-\alpha )}. \]
Taking the supremum over $p\in B_ r$ and absorbing fixed powers of $d^{-1}$, we obtain
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\begin{equation} M_ j(r)\leq Cd^{-1}M_ j(R)^ q, \qquad q=\frac{N(1-\alpha )}{2-\alpha }. \end{equation}
Choose $\alpha _*$ so close to one that $q<1$ for $\alpha >\alpha _*$. Choose $0<r_0<R_0$ with $B_{R_0}\Subset B_0$. Let $r_ k\uparrow R_0$ with $r_{k+1}-r_ k=(R_0-r_0)2^{-k-1}$. Iterating (15.18.48) gives
\[ M_ j(r_0) \leq \prod _{k=0}^{m-1}\left[C(r_{k+1}-r_ k)^{-1}\right]^{q^ k} M_ j(r_ m)^{q^ m}. \]
The infinite product converges because $\sum q^ k(1+k)<\infty $. For each fixed $j$, the final factor tends to one as $m\to \infty $. The resulting bound is independent of $j$.
$\square$
Pome
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