Bicomplex Hardy Classes of Solutions to Higher-Order Vekua Equations

Technical Summary: Bicomplex Hardy Classes of Solutions to Higher-Order Vekua Equations

Problem Statement
This paper addresses the extension of Hardy space theory to solutions of differential equations in the bicomplex number setting. Specifically, it seeks to generalize the boundary behavior results known for classical holomorphic Hardy spaces, generalized analytic functions (solutions to the complex Vekua equation), and higher-order iterated Vekua equations (HOIV) to the realm of bicomplex-valued functions. While Hardy spaces for bicomplex-holomorphic functions and solutions to the first-order bicomplex Vekua equation had been partially explored in prior literature (e.g., [12], [43], [44]), a comprehensive theory for the higher-order generalizations of these equations within the bicomplex framework was lacking. The central problem is to define appropriate bicomplex Hardy classes for these solutions and to prove that they exhibit the same fundamental boundary properties as their complex counterparts: the existence of nontangential boundary values in $L^p$, convergence in the $L^p$ norm, and the existence of distributional boundary values.

Methodology
The author employs a structural approach based on the idempotent representation of bicomplex numbers. The bicomplex algebra $\mathbb{B}$ is decomposed using idempotent elements $p_\pm = \frac{1}{2}(1 \pm ij)$, allowing any bicomplex-valued function $f$ to be written as $f = p_+ f_+ + p_- f_-$. This decomposition reduces problems involving bicomplex functions to problems involving pairs of complex-valued functions.

The methodology proceeds in several stages:

1. Foundational Definitions: The paper defines bicomplex Lebesgue spaces $L^p(D, \mathbb{B})$ and establishes their relationship to complex Lebesgue spaces via the idempotent components. It introduces the bicomplex differential operators $\partial$ and $\bar{\partial}$ and the bicomplex Theodorescu operator $T_B$, which serves as a right-inverse to $\bar{\partial}$.
2. Representation Theorems: The core of the analysis relies on proving representation formulas for solutions to the bicomplex Vekua equation ($\bar{\partial}w = Aw + B\bar{w}$) and its higher-order iterated variants. These representations mirror the "similarity principle" (first kind) and the integral representation (second kind) known in the complex setting.
3. Reduction to Complex Cases: By utilizing the idempotent decomposition, the author demonstrates that a bicomplex function solves a bicomplex Vekua-type equation if and only if its components solve specific complex Vekua-type equations. This allows the author to leverage existing theorems regarding complex generalized analytic functions (specifically results from [15], [25], and [45]) to derive properties for the bicomplex case.
4. Boundary Analysis: Using the established representations and the known boundary behavior of the complex components, the paper proves convergence results for the bicomplex functions.

Key Contributions and Results

• Definition of Bicomplex Hardy Classes: The paper formally defines the bicomplex Vekua-Hardy spaces $H^p_{A,B}(D, \mathbb{B})$ and the bicomplex higher-order iterated Vekua (HOIV) Hardy spaces $V^{n,p}_{A,B}(D, \mathbb{B})$. These spaces consist of solutions to the respective differential equations with finite $H^p$ norms.
• Representation Formulas:
  • First Kind: For the case $B \equiv 0$, solutions are shown to be of the form $w = \phi e^\varphi$, where $\phi$ is bicomplex-holomorphic and $\varphi$ is related to the Theodorescu operator of the coefficient $A$.
  • Second Kind: General solutions are represented as $w = \phi + T_B(Aw + B\bar{w})$.
  • Polyanalytic/Meta-Analytic Structure: For higher-order equations, solutions are shown to admit polynomial representations. Specifically, solutions to the bicomplex meta-analytic equation (where $B=0$) are polynomials in the bicomplexification of the conjugate variable ($\bar{z}$) with coefficients in the corresponding bicomplex Vekua-Hardy spaces. For the general HOIV case, solutions are polynomials in $(\bar{z} + z)$ with coefficients in the bicomplex Vekua-Hardy spaces.
• Boundary Behavior:
  • Nontangential Limits: The paper proves that functions in these bicomplex Hardy classes possess nontangential boundary values $f_{nt} \in L^p(\partial D, \mathbb{B})$ almost everywhere.
  • Norm Convergence: It is shown that these functions converge to their boundary values in the $L^p$ norm as the radius approaches 1.
  • Distributional Boundary Values: The existence of boundary values in the sense of distributions is established for these classes, extending results from [24] and [15] to the bicomplex setting.
• Inclusion Results: The paper demonstrates that membership in a bicomplex Hardy space implies membership in a range of Lebesgue spaces $L^m(D, \mathbb{B})$ for $0 < m < 2p$, extending classical results for holomorphic and generalized analytic functions.

Significance
The paper claims to directly extend the results found in [15] (which dealt with complex-valued solutions to higher-order iterated Vekua equations) into the bicomplex setting. By doing so, it unifies the study of Hardy spaces for holomorphic functions, generalized analytic functions, and polyanalytic functions within the framework of bicomplex analysis. The work provides a rigorous foundation for analyzing the boundary behavior of solutions to bicomplex differential equations, which arise naturally in the study of the stationary Schrödinger equation and other physical models where bicomplex factorization is applicable. The results confirm that the rich boundary theory of classical Hardy spaces is robust enough to accommodate the algebraic complexity of bicomplex numbers and the non-homogeneity of Vekua-type equations.