On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves

Yan Dai; Juan Bory-Reyes; Fuli He

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arXiv · 2608.30634

On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves

Yan Dai·Juan Bory-Reyes·Fuli He

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Abstract

Hyperanalytic functions, in the sense established by the mathematician Avron Douglis, are Douglis algebra-valued functions defined via a hypercomplex structure rather than the standard Cauchy-Riemann equations characteristic of traditional complex analysis. The classes of polyhyperanalytic and meta-hyperanalytic functions represent advanced generalizations of Douglis's analysis. They are employed in the study of partial differential equations and elasticity, extending the concept of the classical holomorphic function through higher-order iterations and non-homogeneous terms. The aim of this work is to find solvability conditions for a fundamental Riemann-type boundary value problem for spaces of poly-hyperanalytic and meta-hyperanalytic functions defined on an open, bounded, simply connected subset of the complex plane, where the boundary need only be a closed d-summable curve. In fractal geometry, d-summability is a geometric property used to define the boundaries of fractal domains, enabling advanced mathematical integration and calculus on complex structures defined by Jenny Harrison and Alec Norton.

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Yan Dai, Juan Bory-Reyes, Fuli He. 2026-08-31. On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves. https://arxiv.org/abs/2608.30634

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