On octonionic Monge-Amp\`ere equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables
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\`ere 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 Cloc1,1-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.
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