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Regular functions of several quaternionic variables and the Cauchy-Fueter complex. (English) Zbl 0966.35088
This remarkable paper contains in his first part a new proof of the generalization of the famous Hartogs theorem in the $$n$$-dimensional quaternionic space $$\mathbb{H}^n$$ $$(n> 1)$$, i.e. a regular function on $$\mathbb{H}^n\setminus K$$ ($$K$$ compact convex subset of $$\mathbb{H}^n$$) can be uniquely regular extended on $$\mathbb{H}^n$$. This proof is based on an idea of L. Ehrenpreis. Because of the non-commutativity of the “partial” quaternionic derivatives ${\partial\over\partial\overline q_1},\dots, {\partial\over\partial\overline q_n}\quad (q_1,\dots, q_n)\in \mathbb{H}^n,$ it is necessary to use matrix techniques. One has to check that the cokernel of a corresponding symbol matrix is torsion-free which again can be obtained when its minors are relatively prime. Furthermore, for the Cauchy-Fueter system ${\partial f_i\over\partial\overline q_i}= g_i\quad (i= 1,2)$ are got (very impressive) necessary and sufficient solvability conditions. At the end new duality theorems are proved and similarities to Sato’s classical results are lined out.

##### MSC:
 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs 30G99 Generalized function theory 58J10 Differential complexes
##### Keywords:
Hartogs theorem; matrix techniques; new duality theorems
Full Text:
##### References:
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