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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
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