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The Hessian equation in quaternionic space

Hichame Amal, Saïd Asserda, Mohamed Barloub

TL;DR

The paper defines an $m$-subharmonic class $\mathcal{SH}_{m}$ in quaternionic space $\mathbb{H}^{n}$ and the quaternionic Hessian operator via the Baston framework, formalizing the quaternionic analogue of Hessian potential theory. It develops the capacity and quasicontinuity theory for $\mathcal{SH}_{m}$, and establishes global a priori $C^{1,1}$ estimates essential for solving Hessian-type Dirichlet problems. The authors prove existence and uniqueness of the homogeneous Dirichlet problem on the unit ball with continuous boundary data and characterize maximal $m$-subharmonic functions by the vanishing of the quaternionic Hessian measure $(\Delta u)^{m}\wedge \beta^{n-m}$. These results extend complex Hessian theory to quaternionic space, advancing quaternionic pluripotential theory and PDE analysis.

Abstract

In this paper, we introduce $m$-subharmonic functions in quaternionic space $\mathbb{H}^{n}$, we define the quaternionic Hessian operator and solve the homogeneous Dirichlet problem for the quaternionic Hessian equation on the unit ball with continuous boundary data.

The Hessian equation in quaternionic space

TL;DR

The paper defines an -subharmonic class in quaternionic space and the quaternionic Hessian operator via the Baston framework, formalizing the quaternionic analogue of Hessian potential theory. It develops the capacity and quasicontinuity theory for , and establishes global a priori estimates essential for solving Hessian-type Dirichlet problems. The authors prove existence and uniqueness of the homogeneous Dirichlet problem on the unit ball with continuous boundary data and characterize maximal -subharmonic functions by the vanishing of the quaternionic Hessian measure . These results extend complex Hessian theory to quaternionic space, advancing quaternionic pluripotential theory and PDE analysis.

Abstract

In this paper, we introduce -subharmonic functions in quaternionic space , we define the quaternionic Hessian operator and solve the homogeneous Dirichlet problem for the quaternionic Hessian equation on the unit ball with continuous boundary data.
Paper Structure (11 sections, 26 theorems, 131 equations)

This paper contains 11 sections, 26 theorems, 131 equations.

Table of Contents

  1. Prelimenaries
  2. Quaternionic linear algebra
  3. Baston operator $\Delta$
  4. The $m$-subharmonic functions in $\mathbb{H}^{n}$
  5. Basic properties of elementary symmetric functions
  6. The $m$-subharmonic functions
  7. Quasicontinuity of $m$-subharmonic functions.
  8. The Dirichlet problem
  9. A priori estimates
  10. Main Theorem
  11. The maximality

Key Result

Proposition 1.3

For a real valued $\mathcal{C}^2$ function $u$, the matrix $[\frac{\partial^2u}{\partial q_j\partial\bar{q}_k}]$ is hyperhermitian.

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3: W3
  • Theorem 1.4: see pages 145, 146 in J
  • Proposition 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Example 1.9
  • Remark 1.10
  • ...and 45 more