2024/04/04 by Yun Shi, Wei Wang, Shi, Yun +3
Computer Science · Mathematics · #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Dirac (video compression format) #FOS: Mathematics #Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Particle physics #Physics
paper · pdf · doi:10.48550/arxiv.2404.03374
openalex publication_date 2024/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A monogenic function of two vector variables is a function annihilated by the operator consisting of two Dirac operators, which are associated to two variables, respectively. We give the explicit form of differential operators in the Dirac complex resolving this operator and prove its ellipticity directly. This open the door to apply the method of several complex variables to investigate this kind of monogenic functions. We prove the Poincaré lemma for this complex, i.e. the non-homogeneous equations are solvable under the compatibility condition by solving the associated Hodge Laplacian equations of fourth order. As corollaries, we establish the Bochner--Martinelli integral representation formula for this differential operator and the Hartogs' extension phenomenon for monogenic functions. We also apply abstract duality theorem to the Dirac complex to obtain the generalization of Malgrange's vanishing theorem and establish the Hartogs--Bochner extension phenomenon for monogenic functions under the moment condition.