On octonionic Hessian complex and Hartogs--Bochner extension of octonionic pluriharmonic functions in two octonionic variables

Abstract

In this paper, we construct the octonionic Hessian complex, which is the octonionic version of the $\partial \bar{\partial }$-complex. By proving its ellipticity, we can solve the non-homogeneous octonionic Hessian equations under a compatibility condition. This is the octonionic version of the $\partial \bar{\partial }$-lemma. We also obtain the boundary version of octonionic Hessian operator and introduce the notion of a tangentially octonionic pluriharmonic function, which allows us to establish the Hartogs--Bochner extension for tangentially octonionic pluriharmonic functions on the boundary of a domain with connected complement. This solves the octonionic version of Poincaré's problem for pluriharmonic extension on complex space. The main difficulty of this construction is the lack of associativity in octonionic case. However, we can overcome it by careful use of weak form of associativity including Moufang identities and alternativity.