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Uniform estimates: from Yau to Kolodziej

Vincent Guedj, Chinh H. Lu

TL;DR

The paper develops a unified, envelope-based route to uniform a priori bounds for complex Monge–Ampère equations and related nonlinear PDEs on compact Kähler (and Hermitian) manifolds. It proves that oscillation of solutions is controlled by Luxembourg norms of the density $f$ under weights satisfying Condition (K) (Theorem A), and extends the framework to equations governed by determinantal majorization (Theorem B), with bounds independent of complex structure. The approach reduces general degenerate cases to Yau’s classical setting via $ω$-psh envelopes and proves robustness across families of equations and even in the Hermitian context. It further investigates the sharpness of the weight conditions through radial models and provides an alternative proof path, broadening the applicability to Dirichlet problems and other quasi-subharmonic settings.

Abstract

In this note we provide a new and efficient approach to uniform estimates for solutions to complex Monge-Ampere equations, as well as for solutions to geometric PDE's that satisfy a determinantal majorization.

Uniform estimates: from Yau to Kolodziej

TL;DR

The paper develops a unified, envelope-based route to uniform a priori bounds for complex Monge–Ampère equations and related nonlinear PDEs on compact Kähler (and Hermitian) manifolds. It proves that oscillation of solutions is controlled by Luxembourg norms of the density under weights satisfying Condition (K) (Theorem A), and extends the framework to equations governed by determinantal majorization (Theorem B), with bounds independent of complex structure. The approach reduces general degenerate cases to Yau’s classical setting via -psh envelopes and proves robustness across families of equations and even in the Hermitian context. It further investigates the sharpness of the weight conditions through radial models and provides an alternative proof path, broadening the applicability to Dirichlet problems and other quasi-subharmonic settings.

Abstract

In this note we provide a new and efficient approach to uniform estimates for solutions to complex Monge-Ampere equations, as well as for solutions to geometric PDE's that satisfy a determinantal majorization.

Paper Structure

This paper contains 15 sections, 6 theorems, 70 equations.

Table of Contents

  1. Basic facts on quasi-plurisubharmonic functions
  2. Skoda-Zeriahi theorem
  3. The $\omega$-psh envelope
  4. Condition (K)
  5. Various approaches to uniform estimates
  6. From Yau to Kołodziej
  7. Proof of Theorem A
  8. Radial examples
  9. Other nonlinear equations
  10. Quasi-subharmonic functions
  11. $\omega$-subharmonic functions
  12. The set $\Gamma(X,\omega)$
  13. Determinantal majorization
  14. Proof of Theorem B
  15. Alternative proof

Key Result

Theorem 1.2

Fix $A>0$ and $\omega$ a Kähler form such that $\int_X \omega \wedge \omega_X^{n-1} \leq A$. There exists $\alpha=\alpha(n,A)>0$ such that for all $\varphi \in PSH_0(X,\omega)$, where $C=C(\alpha,n,A)>0$ is independent of $\omega$ and $\varphi$.

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Example 1.6
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 11 more