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

William L. Blair

TL;DR

The paper develops a comprehensive bicomplex Hardy space framework for first-order elliptic PDEs, beginning with a bridge to complex Beltrami, conjugate Beltrami, and Vekua equations, then embedding these into bicomplex analysis via idempotent decomposition. It defines and analyzes H^p_{Bel,\mu}(\mathbb{D},\mathbb{B}), H^p_{conj,\mu}(\mathbb{D},\mathbb{B}), and higher-order HOIB variants, showing their norms and boundary behavior reduce to corresponding complex-component Hardy spaces. By extending to general first-order elliptic equations and establishing the bicomplex Schwarz and Dirichlet boundary-value problems, the work provides explicit representation formulas and solvability criteria in the bicomplex setting. This advances boundary-value theory and boundary convergence for bicomplex-differential equations, with potential applications in bicomplex-analytic models of physical phenomena and in generalized Hardy-space frameworks. The main achievement is a cohesive, representation-based approach that preserves classical Hardy-space properties while enabling bicomplex PDE analysis through componentwise complex analogues.

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.

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

TL;DR

The paper develops a comprehensive bicomplex Hardy space framework for first-order elliptic PDEs, beginning with a bridge to complex Beltrami, conjugate Beltrami, and Vekua equations, then embedding these into bicomplex analysis via idempotent decomposition. It defines and analyzes H^p_{Bel,\mu}(\mathbb{D},\mathbb{B}), H^p_{conj,\mu}(\mathbb{D},\mathbb{B}), and higher-order HOIB variants, showing their norms and boundary behavior reduce to corresponding complex-component Hardy spaces. By extending to general first-order elliptic equations and establishing the bicomplex Schwarz and Dirichlet boundary-value problems, the work provides explicit representation formulas and solvability criteria in the bicomplex setting. This advances boundary-value theory and boundary convergence for bicomplex-differential equations, with potential applications in bicomplex-analytic models of physical phenomena and in generalized Hardy-space frameworks. The main achievement is a cohesive, representation-based approach that preserves classical Hardy-space properties while enabling bicomplex PDE analysis through componentwise complex analogues.

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.

Paper Structure

This paper contains 31 sections, 37 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Complex Origins
  4. Holomorphic Functions
  5. Beltrami Equations
  6. Conjugate Beltrami Equations
  7. Connection to Vekua Equations
  8. General First Order Elliptic Equations
  9. Bicomplex Numbers
  10. Basics of Bicomplex Numbers
  11. Bicomplex Holomorphic Functions
  12. Bicomplex Vekua Equations
  13. Bicomplex Hardy Spaces
  14. The Schwarz and Dirichlet Boundary Value Problems
  15. The Beltrami Equation
  16. ...and 16 more sections

Key Result

Theorem 2.2

A function $f \in H^p(\mathbb{D})$, $0 < p < \infty$, has nontangential boundary values $f_{nt} \in L^p(\partial \mathbb{D})$ at almost every point of $\partial \mathbb{D}$, and

Theorems & Definitions (97)

  • Definition 2.1
  • Theorem 2.2: DurenKoosisRepRudinCRBAF
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Theorems 6.1.2, 6.1.3, 6.1.4, and 6.1.5 KlimBook, Theorems 1, 2, 3, and 4 Klim2016, Theorem 4.1, Corollary 4.4 quasiHardy, Zins
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: Theorems 1, 2, 3, and 4 Klim2016, Proposition 4.3.1 conjbel
  • Definition 2.9
  • Definition 2.10
  • ...and 87 more