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Degenerate quarternionic Monge-Ampère equations in weighted energy classes

Genglong Lin

TL;DR

The paper extends quaternionic pluripotential theory to degenerate Monge-Ampère equations by introducing weighted energy classes $\mathcal{E}_\chi(\Omega)$ and systematically establishing when the operator $$(\partial\partial_J u)^n$$ is well-defined on these classes. It proves inclusions $\mathcal{E}_p(\Omega)\subset \mathcal{E}(\Omega)$ and $\mathcal{E}_\chi(\Omega)\subset \mathcal{E}(\Omega)$, proves a plurifine property within $\mathcal{E}(\Omega)$, and derives a mass concentration theorem for the quaternionic plurisubharmonic envelope $P(f)$. A central contribution is a Guedj-Zeriahi-type characterization of the finite energy range of the quaternionic Monge-Ampère operator, giving equivalent conditions for when a measure $\mu$ is the Monge-Ampère image of some $\varphi\in\mathcal{E}_\chi(\Omega)$. Together, these results provide a robust framework for solving Dirichlet problems with energy data in the quaternionic setting and extend complex-analytic energy techniques to hypercomplex spaces.

Abstract

In this paper, we consider degenerate quaternionic Monge-Ampère equations in weighted energy class $\mathcal{E}_χ(Ω)$ where $Ω$ is a quarternionic domain in $\mathbb{H}^n$ and $χ$ is a weight function which satisfies some natural conditions. Firstly we prove that the quaternionic Monge-Ampère operator is well-defined for functions in $\mathcal{E}_χ(Ω)$, in particular $\mathcal{E}_p(Ω),p>0$. Secondly, we prove that fine property holds in the Cegrell type class $\mathcal{E}(Ω)$. As an application, we prove a mass concentration theorem for the quarternionic plurisubharmonic envelope. In the study of complex Monge-Ampère equation, characterization of finite energy range of complex Monge-Ampère operator was a central problem which aroused the interest of experts in the subject. As a quaternionic analogue, we prove a theorem which explicitly characterizes the finite energy range of \emph{quaternionic} Monge-Ampère operator in the end.

Degenerate quarternionic Monge-Ampère equations in weighted energy classes

TL;DR

The paper extends quaternionic pluripotential theory to degenerate Monge-Ampère equations by introducing weighted energy classes and systematically establishing when the operator is well-defined on these classes. It proves inclusions and , proves a plurifine property within , and derives a mass concentration theorem for the quaternionic plurisubharmonic envelope . A central contribution is a Guedj-Zeriahi-type characterization of the finite energy range of the quaternionic Monge-Ampère operator, giving equivalent conditions for when a measure is the Monge-Ampère image of some . Together, these results provide a robust framework for solving Dirichlet problems with energy data in the quaternionic setting and extend complex-analytic energy techniques to hypercomplex spaces.

Abstract

In this paper, we consider degenerate quaternionic Monge-Ampère equations in weighted energy class where is a quarternionic domain in and is a weight function which satisfies some natural conditions. Firstly we prove that the quaternionic Monge-Ampère operator is well-defined for functions in , in particular . Secondly, we prove that fine property holds in the Cegrell type class . As an application, we prove a mass concentration theorem for the quarternionic plurisubharmonic envelope. In the study of complex Monge-Ampère equation, characterization of finite energy range of complex Monge-Ampère operator was a central problem which aroused the interest of experts in the subject. As a quaternionic analogue, we prove a theorem which explicitly characterizes the finite energy range of \emph{quaternionic} Monge-Ampère operator in the end.
Paper Structure (4 sections, 21 theorems, 88 equations)

This paper contains 4 sections, 21 theorems, 88 equations.

Table of Contents

  1. introduction
  2. Preliminary
  3. Cegrell type classes
  4. Weighted energy classes

Key Result

Theorem 1.1

(Proposition energy class in cegrell class) In particular, the quaternionic Monge-Ampère operator $(\partial\partial_J u)^n$ is well-defined for any $u\in\mathcal{E}_{\chi}(\Omega)$ and $-\chi(u)\in L^1((\partial\partial_J u)^n)$. Moreover, $(\partial\partial_J u)^n$ puts no mass on quaternionic polar subsets.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • ...and 30 more