Lemma 14.17.119E4D601D907

Proof. Work with $v=u/(2\pi )$ and write $Q=Q_ u$ for the matrix in (14.13.1). By proposition 14.11.1, theorem 14.14.1, proposition 14.16.1, the function $v$, its gradient and its complex Hessian are uniformly bounded, and so are the constants $b$. Hence the eigenvalues of $Q$ are uniformly bounded above, and by the equation its determinant is uniformly bounded below. Every eigenvalue is therefore also bounded below by a positive constant. Choose fixed numbers $0<a<b_0$ such that $aI\leq Q\leq b_0I$ in unitary frames for every solution.

The function $\Delta _\gamma v=\operatorname {tr}_\gamma (\mathrm i\partial \bar\partial v)$ is uniformly bounded, and $\Delta _\gamma $ is a fixed uniformly elliptic operator. Its interior $W^{2,p}$ estimates, applied on a finite cover, bound $v$ in $W^{2,p}$ for every finite $p$. Choose $p>2n$ and $0<\alpha <1-2n/p$; Sobolev embedding gives a uniform $C^{1,\alpha }$ bound. Thus the matrix $R(x)=Q_0(x)+Z_ x(\partial v(x))$ is uniformly bounded in $C^\alpha $.

On a coordinate ball, let $\mathcal H_ x(A)$ denote the Hermitian matrix of $\mathrm i\partial \bar\partial v$ at $x$ in a fixed smooth unitary frame, expressed as a function of the real Hessian $A=D^2_{\mathbb R}v(x)$. It is a real linear function of $A$ depending smoothly on $x$. If a real symmetric matrix $N$ is nonnegative, then $\mathcal H_ x(N)\geq 0$, and its trace is comparable to $\operatorname {tr}_{\mathbb R}N$ with uniform positive constants on the ball. This follows by evaluating on a vector and its rotation by $\mathrm i$, or from the coordinate formula for $v_{i\bar j}$.

Freeze the function $\partial v(x)$ already obtained in $R(x)$, and define for all real symmetric matrices $A$

The infimum is over a fixed compact set of positive Hermitian matrices, and each expression in braces is an affine function of $A$, so $\widetilde\Phi $ is concave in $A$. If $R(x)+P_ n(\mathcal H_ x(A))$ lies in that compact set, then $\widetilde\Phi (A,x)$ equals its logarithmic determinant: each expression in braces is a tangent affine function of the concave logarithmic determinant, and the tangent at that matrix itself attains equality.

For $N\geq 0$, each of these affine functions increases by

when $A$ is replaced by $A+N$. Since $P_ n$ preserves positivity and trace, this increment lies between fixed positive multiples of $\operatorname {tr}_{\mathbb R}N$, uniformly in $B$ and $x$. Taking infima preserves both inequalities, so $\widetilde\Phi $ is uniformly elliptic on all real symmetric matrices.

Smoothness of $\mathcal H_ x$ and the uniform $C^\alpha $ bound for $R$ also give

These are the hypotheses of the local Evans–Krylov estimate for concave uniformly elliptic equations with Hölder dependence on $x$, in the formulation given in section 25.8. The value $\widetilde\Phi (0,x)$ is uniformly bounded, by the compactness of the set of matrices $B$ and the bound for $R$. By the choice of $a$ and $b_0$, the given smooth function solves

on the ball. The Evans–Krylov estimate, applied on smaller balls of a finite cover, yields a uniform $C^{2,\beta }$ bound for some $\beta >0$.

Differentiating the original matrix equation gives linear uniformly elliptic equations for the first derivatives of $v$, whose coefficients, including the first-order coefficients coming from $Z$, are now Hölder continuous. Local Schauder estimates give $C^{3,\beta }$ bounds, and differentiation and induction give all higher bounds. The finite coordinate cover makes the bounds global. The only regularity results used here are the standard $W^{2,p}$ estimates for the Laplacian, the Evans–Krylov estimate for concave equations, and the Schauder estimates. $\square$