<p>We prove an Alexandrov–Bakelman–Pucci type estimate, which involves the integral of the determinant of the complex Hessian over a certain subset. It improves the classical ABP estimate adapted (by inequality <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$2^{2n}|\det (u_{i\bar{j}})|^2\ge |\det (\nabla ^2u)|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <mo stretchy="false">|</mo> <mo movablelimits="true">det</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow> <mi>i</mi> <mover accent="true"> <mrow> <mi>j</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo>≥</mo> <mrow> <mo stretchy="false">|</mo> <mo movablelimits="true">det</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>) to complex setting. We give an application of it to derive sharp gradient estimates for complex Monge–Ampère equations. The approach is based on the De Giorgi iteration method developed by Guo–Phong–Tong for equations of complex Monge–Ampère type.</p>

Complex Alexandrov–Bakelman–Pucci Estimate and Its Applications

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2026年5月5日

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Junbang Liu

摘要

We prove an Alexandrov–Bakelman–Pucci type estimate, which involves the integral of the determinant of the complex Hessian over a certain subset. It improves the classical ABP estimate adapted (by inequality $2^{2n}|\det (u_{i\bar{j}})|^2\ge |\det (\nabla ^2u)|$ $2^{2 n} | det (u_{i \bar{j}}) |^{2} \geq | det (\nabla^{2} u) |$ ) to complex setting. We give an application of it to derive sharp gradient estimates for complex Monge–Ampère equations. The approach is based on the De Giorgi iteration method developed by Guo–Phong–Tong for equations of complex Monge–Ampère type.