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On the sharp $L^2$-estimates of Skoda division theorem

Masakazu Takakura

TL;DR

The paper develops a sharp $L^2$-division theorem of Skoda type on weakly Kähler manifolds by combining refined $ar{ ext{d}}$-estimates with carefully chosen triples $(C,D,S)$ in $ obreak \mathcal{G}_{oldsymbol{\Phi}}$. It proves Theorem A, deriving optimal division bounds for surjective bundle maps and extending to vector bundles, while establishing a priori estimates that underpin sharpness. A key contribution is showing that the sharp $ obreak L^2$-division property characterizes plurisubharmonicity for $C^2$ weights (Theorem B) and that this sharp division framework yields a new route to Guan–Zhou's sharp $L^2$-extension theorem. The results deepen the link between $L^2$ solvability, curvature positivity, and convexity properties, and provide new tools via multiplier-ideal techniques and an ellipsoidal mean-value perspective for plurisubharmonicity.

Abstract

In this paper, we prove a Skoda type division theorem with sharp $L^2$-estimate. Furthermore, using this estimate, we provide new characterizations of plurisubharmonic functions. We also explain that the sharp $L^2$-division theorem leads the Guan-Zhou's sharp $L^2$-estimate for extension theorem.

On the sharp $L^2$-estimates of Skoda division theorem

TL;DR

The paper develops a sharp -division theorem of Skoda type on weakly Kähler manifolds by combining refined -estimates with carefully chosen triples in . It proves Theorem A, deriving optimal division bounds for surjective bundle maps and extending to vector bundles, while establishing a priori estimates that underpin sharpness. A key contribution is showing that the sharp -division property characterizes plurisubharmonicity for weights (Theorem B) and that this sharp division framework yields a new route to Guan–Zhou's sharp -extension theorem. The results deepen the link between solvability, curvature positivity, and convexity properties, and provide new tools via multiplier-ideal techniques and an ellipsoidal mean-value perspective for plurisubharmonicity.

Abstract

In this paper, we prove a Skoda type division theorem with sharp -estimate. Furthermore, using this estimate, we provide new characterizations of plurisubharmonic functions. We also explain that the sharp -division theorem leads the Guan-Zhou's sharp -estimate for extension theorem.
Paper Structure (15 sections, 15 theorems, 94 equations)

This paper contains 15 sections, 15 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Theorem
  4. Positivity of chern curvature forms
  5. Refinement of Hörmander's estimate
  6. Sharp $L^2$-estimate of Skoda-type division theorem
  7. Proof of Theorem A
  8. Examples
  9. Psuedoconvex domain case
  10. Division problem for holomorphic sections on vector bundles
  11. Characterizing plurisubharmonicity via the sharp $L^2$-division property
  12. Correspondences between the solvability of analytic equations and the convexities
  13. Proof of Theorem B
  14. A proof of sharp $L^2$-extension theorem
  15. Another characterization of plurisubharmonicity via the sharp $L^2$-division

Key Result

Theorem 1.1

Let $\Omega$ be a psuedoconvex domain in $\mathbb{C}^n$ and $\phi$ be a plurisubharmonic function on $\Omega$. Let $g = (g_1,\dots,g_r)$ be non-zero $r$-tuple holomorphic functions and put $q = \min(n,r-1)$. Then for any holomorphic function $f$ satisfying for lebesgue measure $d\lambda$, there exists a tuple of holomorphic function $F = (F_1,\dots,F_r)$ satisfying

Theorems & Definitions (39)

  • Theorem 1.1: Skoda72
  • Example 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Example 1.7
  • Definition 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 29 more