Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables

Wei Wang

TL;DR

This work extends pluripotential theory to octonionic plurisubharmonic functions of two variables, navigating non-associativity by exploiting a weak associativity framework. It develops a full toolkit: an integration-by-parts formula for the mixed octonionic Monge-Ampère operator, uniform estimates, and a comparison principle; identifies a fundamental solution and Lelong numbers, and studies the Dirichlet problem with $C_{loc}^{1,1}$ regularity via Bedford–Taylor methods augmented by weighted automorphism transformations. The paper also defines octonionic relative extremal functions and capacity, establishing their basic properties and connections to regular compact sets; finally, it proves quasicontinuity for locally bounded OPSH functions. Together, these results lay a robust octonionic pluripotential framework with parallels to complex and quaternionic theories, enabling further development of octonionic potential theory and its geometric applications.

Abstract

Several aspects of pluripotential theory are generalized to octonionic plurisubharmonic (OPSH) functions of two variables. We prove the comparison principle for continuous OPSH functions and the quasicontinuity of locally bounded ones. An important tool is a formula of integration by parts for mixed octonionic Monge-Ampère operator. Various useful properties of octonionic relative extremal functions and octonionic capacity are established. The main difficulty is the non-associativity of octonions. However, some weak form of associativity can be used to covercome this difficulty. Another important ingredient in pluripotential theory is the solution to the Dirichlet problem for the homogeneous octonionic Monge-Ampère equation on the unit ball, for which we show the $C_{loc}^{1,1}$-regularity by applying Bedford-Taylor's method. The obstacle to do so is that an OPSH function is usually not OPSH under automorphisms of the unit ball. This issue can be solved by finding a weighted transformation formula of OPSH functions.

On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables

TL;DR

This work extends pluripotential theory to octonionic plurisubharmonic functions of two variables, navigating non-associativity by exploiting a weak associativity framework. It develops a full toolkit: an integration-by-parts formula for the mixed octonionic Monge-Ampère operator, uniform estimates, and a comparison principle; identifies a fundamental solution and Lelong numbers, and studies the Dirichlet problem with regularity via Bedford–Taylor methods augmented by weighted automorphism transformations. The paper also defines octonionic relative extremal functions and capacity, establishing their basic properties and connections to regular compact sets; finally, it proves quasicontinuity for locally bounded OPSH functions. Together, these results lay a robust octonionic pluripotential framework with parallels to complex and quaternionic theories, enabling further development of octonionic potential theory and its geometric applications.

Abstract

Several aspects of pluripotential theory are generalized to octonionic plurisubharmonic (OPSH) functions of two variables. We prove the comparison principle for continuous OPSH functions and the quasicontinuity of locally bounded ones. An important tool is a formula of integration by parts for mixed octonionic Monge-Ampère operator. Various useful properties of octonionic relative extremal functions and octonionic capacity are established. The main difficulty is the non-associativity of octonions. However, some weak form of associativity can be used to covercome this difficulty. Another important ingredient in pluripotential theory is the solution to the Dirichlet problem for the homogeneous octonionic Monge-Ampère equation on the unit ball, for which we show the -regularity by applying Bedford-Taylor's method. The obstacle to do so is that an OPSH function is usually not OPSH under automorphisms of the unit ball. This issue can be solved by finding a weighted transformation formula of OPSH functions.
Paper Structure (17 sections, 40 theorems, 188 equations)

This paper contains 17 sections, 40 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Octonionic linear algebra, $OPSH$ functions and octonionic closed positive currents
  3. Octonions and octonionic Hermitian $(2 \times 2)$-matrices
  4. $OPSH$ functions and octonionic closed positive currents
  5. Integration by parts
  6. Octonionic mixed Monge-Ampère measure and the comparison principle
  7. A uniform estimate
  8. The comparison principle
  9. The fundamental solution of the octonionic Monge-Ampère equation and the Lelong number for a closed positive current
  10. The Dirichlet problem for the homogeneous octonionic Monge-Ampère equation
  11. Weighted transformation formula of $OPSH$ functions under automorphisms of the unit ball
  12. The regularity of the Perron-Bremermann function
  13. Octonionic relative extremal function and octonionic capacity
  14. Octonionic relative extremal function
  15. Octonionic capacity
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain and let $u,v\in OPSH(\Omega)\cap C(\Omega)$. If $\{u<v\} \Subset \Omega$, then we have

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.1
  • Lemma 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.2
  • Proposition 2.4
  • ...and 59 more