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A bridge connecting convex analysis and complex analysis and $L^2$-estimate of $d$ and $\bar\partial$

Fusheng Deng, Jinjin Hu, Weiwen Jiang, Xiangsen Qin

TL;DR

This work builds a principled bridge between convex analysis and complex analysis by transferring problems from convex domains in $\mathbb{R}^n$ to rotation-invariant settings on Reinhardt domains via tube domains and the exponential map. The authors develop and apply $L^2$-estimates for the $\bar{\partial}$-equation and the $d$-operator within this framework, obtaining both sharp extension results and curvature-positivity conclusions for direct-image bundles. Key contributions include a Berndtsson-style $L^2$-estimate on tube-type domains, an optimal $L^2$-extension theorem with explicit constants, and Nakano positivity results for direct images in families of circular/Reinhardt domains, with convex-domain corollaries. The results illuminate how complex-analytic techniques yield insights and tools for convex-analytic problems, potentially enabling new Brunn-Minkowski-type inequalities in real settings via complex-analytic methods.

Abstract

We propose a way to connect complex analysis and convex analysis. As applications, we derive some results about $L^2$-estimate for $d$-equation and prove some curvature positivity related to convex analysis from well known $L^2$-estimate for $\bar\partial$-equation or the results we prove in complex analysis.

A bridge connecting convex analysis and complex analysis and $L^2$-estimate of $d$ and $\bar\partial$

TL;DR

This work builds a principled bridge between convex analysis and complex analysis by transferring problems from convex domains in to rotation-invariant settings on Reinhardt domains via tube domains and the exponential map. The authors develop and apply -estimates for the -equation and the -operator within this framework, obtaining both sharp extension results and curvature-positivity conclusions for direct-image bundles. Key contributions include a Berndtsson-style -estimate on tube-type domains, an optimal -extension theorem with explicit constants, and Nakano positivity results for direct images in families of circular/Reinhardt domains, with convex-domain corollaries. The results illuminate how complex-analytic techniques yield insights and tools for convex-analytic problems, potentially enabling new Brunn-Minkowski-type inequalities in real settings via complex-analytic methods.

Abstract

We propose a way to connect complex analysis and convex analysis. As applications, we derive some results about -estimate for -equation and prove some curvature positivity related to convex analysis from well known -estimate for -equation or the results we prove in complex analysis.
Paper Structure (13 sections, 18 theorems, 135 equations)

This paper contains 13 sections, 18 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Regularized max function
  5. Curvature positivity of Hermitian holomorphic vector bundles
  6. $L^2$-estimates in complex analysis
  7. Curvature positivity of real vector bundles
  8. Some basic knowledge from measure theory
  9. The proof of Theorem \ref{['thm:Ber']}
  10. The proof of Theorem \ref{['thm:optimal']}
  11. The proof of Theorem \ref{['thm:Cor']}
  12. The proof of Theorem \ref{['thm(extend):curvartue positive circular domains']}
  13. The proof of Theorem \ref{['qin:convex']}

Key Result

Theorem 1.1

Let $D\subset \mathbb{R}_x^n$ be a convex domain, $V\subset \mathbb{C}_\tau^m$ be a pseudoconvex domain, and let $W_z:=D+i \mathbb{R}_y^n\subset \mathbb{C}_z^n.$ Let $\varphi(\tau,z),\ \psi(\tau, z)$ be plurisubharmonic functions on $V\times W,$ which are independent of $\operatorname{Im}(z)$ (the i provided the right hand side is finite, where $\left(\psi^{j\bar{k}}\right)_{n\times n}:=\left(\fra

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2: I21
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: Dem
  • Definition 2.1
  • Lemma 2.2: see Dem
  • Lemma 2.3: Ber95
  • ...and 20 more