Bicomplex Hardy Classes of Solutions to Beltrami Equations and the Schwarz Boundary Value Problem

TL;DR

This paper introduces bicomplex Hardy classes of solutions to Beltrami equations, extending classical Hardy space results to bicomplex functions and solving related boundary value problems.

Contribution

It generalizes Hardy space theory to bicomplex-valued functions and provides solvability and explicit formulas for boundary value problems in this setting.

Findings

Bicomplex Hardy classes of solutions are defined and characterized.

Boundary behavior of bicomplex solutions aligns with classical Hardy spaces.

Schwarz and Dirichlet problems for bicomplex Beltrami equations are solvable with explicit formulas.

Abstract

We define Hardy classes of bicomplex-valued functions on the complex unit disk which solve bicomplex versions of the Beltrami and related equations. Using representations in terms of their complex-valued counterparts, we show these bicomplex-valued functions recover the boundary behavior associated with the classic holomorphic Hardy spaces. This work generalizes known results for complex-valued functions and continues recent work in the setting of bicomplex analogues of Hardy spaces of both holomorphic and generalized analytic functions. Also, we show Schwarz and Dirichlet boundary value problems associated with the bicomplex Beltrami equation are solvable and provide solution formulas.