O -Regular Mappings on C ( C ): A Structured Operator–Theoretic Framework

Ji Eun Kim ()
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Ji Eun Kim: Department of Mathematics, Dongguk University, WISE, Gyeongju 38066, Republic of Korea

Mathematics, 2025, vol. 13, issue 20, 1-17

Abstract: Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model C ( C ) ≅ C 4 has not been systematically developed. Purpose and Aims. This paper develops such a calculus for O -regular mappings on C ( C ) and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing O -regularity; (ii) identification of a canonical first derivative g ′ ( z ) = ∂ x 0 g ( z ) ; and (iii) a Stokes-driven boundary annihilation law ∫ ∂ Ω τ g = 0 for a canonical 7-form τ . On (pseudo)convex domains, ∂ ¯ -methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on C ( C ) .

Keywords: O -regular functions; Dirac-type operators; coupled Cauchy–Riemann system; Cauchy–Stokes boundary identity; Borel–Pompeiu; ∂ ¯ methods; several complex variables; hypercomplex/commutative algebras; boundary integral methods (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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