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

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.