AI 中文总结
本文构造八元数Hessian复形，克服八元数非结合性困难，求解非齐次Hessian方程，建立切向八元数多重调和函数的Hartogs–Bochner延拓，解决了庞加莱问题的八元数版本。
AI 中文摘要
本文构造了八元数Hessian复形，它是∂\bar∂-复形的八元数版本。通过证明其椭圆性，我们可在相容条件下求解非齐次八元数Hessian方程，这是∂\bar∂-引理的八元数版本。我们还得到了八元数Hessian算子的边界版本，引入了切向八元数多重调和函数的概念，由此在补集连通的区域边界上建立了切向八元数多重调和函数的Hartogs–Bochner延拓，解决了复空间上多重调和延拓的庞加莱问题的八元数版本。该构造的主要困难在于八元数情形缺乏结合性，但我们可通过谨慎运用含Moufang恒等式与交错性的弱结合性克服这一点。
英文摘要
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.
