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

TL;DR

This paper establishes a novel connection between convex analysis and complex analysis, deriving new $L^2$-estimates for differential operators and exploring curvature positivity through this interdisciplinary approach.

Contribution

It introduces a new framework linking convex and complex analysis, leading to novel $L^2$-estimates and curvature results in complex analysis.

Findings

Derived $L^2$-estimates for the $d$-equation

Proved curvature positivity results from convex analysis

Connected complex and convex analysis through a unified approach

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.