On octonionic Monge-Amp\`{e}re equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables

TL;DR

This paper extends pluripotential theory to octonionic plurisubharmonic functions of two variables, establishing fundamental principles, properties, and regularity results despite the challenges posed by octonion non-associativity.

Contribution

It introduces a generalized pluripotential framework for octonionic functions, proves the comparison principle, quasicontinuity, and regularity of solutions to the octonionic Monge-Ampère equation.

Findings

Proved comparison principle for continuous OPSH functions

Established $C_{loc}^{1,1}$-regularity for solutions to the Dirichlet problem

Developed a weighted transformation formula for OPSH functions

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-Amp\`{e}re 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-Amp\`ere 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.