Vekua Equation: Generalized Analytic Framework

• Vekua equation is a broad class of first-order PDEs that generalizes analytic function theory by coupling a function with its conjugate.
• It underpins diverse frameworks, including Clifford-valued and bicomplex settings, and extends classical spaces like Bergman and Hardy spaces.
• Its applications span spectral theory, conductivity, and quantum mechanics, highlighting its versatility in solving complex physical and inverse problems.

A Vekua equation is a broad class of first-order partial differential equations (PDEs) fundamentally generalizing analytic function theory to encompass diverse analytic and pseudoanalytic frameworks, scalar and Clifford-valued function theories, elliptic and nonelliptic systems, and applications in mathematical physics. Originating from the monograph of I. N. Vekua, these equations serve as the foundational structure connecting generalized analytic, pseudoanalytic, and monogenic function theories across Euclidean spaces of various dimensions and algebraic structures.

1. Definition and General Algebraic Structure

A prototypical Vekua equation in the complex plane on a domain Ω ⊂ ℂ is

$$\partial_{\bar z} W(z) = a(z)\,W(z) + b(z)\,\overline{W(z)},\qquad z\in\Omega,$$

where $\partial_{\bar z}$ is the Cauchy–Riemann operator, $a(z)$ and $b(z)$ are coefficient functions that may be complex-, matrix-, Clifford-, or bicomplex-valued depending on context, and $\overline{W}$ could be complex or algebraic conjugation according to the setting. This formulation recovers holomorphic function theory for $a=b=0$, but more generally couples $W$ and its conjugate, inducing a real-linear (not complex-linear) structure and anisotropy in the space of solutions (Campos et al., 2018).

Extended forms include higher-dimensional and algebra-valued settings:

• Clifford-valued Vekua equations: $D w = \alpha C[w] + w \beta$ with $D$ the Moisil–Teodorescu operator, $\alpha,\beta$ bounded Clifford-valued functions, and $C$ Clifford conjugation (Delgado, 2023, Delgado, 24 Jan 2026).
• Bicomplex/Biquaternionic cases: $\partial_{\bar z} W = aW + b\widetilde{W}$, where $W$ is bicomplex and $\widetilde{W}$ the relevant conjugate (Vicente-Benítez, 2023, Campos et al., 2012, Vicente-Benítez, 2024).
• On compact Lie groups: $P u = L u - A u - B \bar u$, $L$ a left-invariant differential operator, $A,B$ scalars or matrices, and group-Fourier analysis replacing classical variables (Kirilov et al., 2023, Moraes, 2021).

2. Solvability and Global Hypoellipticity

Vekua-type operators present nontrivial mapping and regularity theories because of their coupling of $W$ and its conjugate. On the torus $T^n$, the Fourier representation reduces the problem to a countable system of $2 \times 2$ (or $d \times d$ matrix-valued) linear systems for each discrete frequency, whose solvability and regularity hinge on the nonvanishing of the associated "discriminant" determinant

$$\Delta_\xi = (\sigma_L(\xi) - A)\,(\overline{\sigma_L(-\xi)} - \overline{A}) - |B|^2\,,$$

with $\sigma_L(\xi)$ the symbol of the differential operator (Kirilov et al., 2023).

The main solvability theorem states that $P$ is solvable if and only if there is a lower (Diophantine-type) bound for $|\Delta_\xi|$ outside a finite set; global hypoellipticity is equivalent to solvability. For elliptic $L$, such lower bounds are automatic, and for equations on compact Lie groups, solvability and hypoellipticity are controlled by spectral gaps for group representations and the parameters $A,B$ (Moraes, 2021).

3. Function Spaces: Bergman, Hardy, and Hodge Theory Extensions

The space of $L^p$- or $C^1$-solutions to a Vekua equation on a domain forms fundamentally new function spaces:

• Bergman spaces ($A^2$-spaces): weak solutions in $L^2$ for the Vekua operator with reproducing kernel properties, generalized projection formulas, and explicit series representations in terms of orthonormal bases ("formal powers") (Campos et al., 2018, Vicente-Benítez, 2024, Vicente-Benítez, 2023, Vicente-Benítez, 2024).
• Hardy spaces: Adapted $H^p$ spaces for solutions of Vekua or higher-order Vekua-type equations, with nontangential boundary limits, distributional boundary values, and second-kind integral representations (Blair, 7 May 2025).
• Hodge decompositions: $L^2$ solutions of Clifford-valued Vekua equations admit orthogonal direct sum decompositions, leading to explicit connections to Schrödinger-type operators and spectral theory (Delgado, 2023).

These spaces are real Hilbert spaces (not complex unless $b=0$), often closed and reflexive Banach spaces with bounded point evaluation functionals due to interior Hölder or Sobolev regularity (Campos et al., 2018, Vicente-Benítez, 2024, Delgado, 2023, Vicente-Benítez, 2024).

4. Kernel Theory and Integral Representations

The Bergman kernel for the Vekua equation is a generalization of the classical holomorphic case and facilitates explicit evaluation, integral representation, and projection formulas. For $W \in H^2$,

$W(\zeta) = \int_\Omega B(a,\zeta;z) W(z)\,dA(z),$

where $B$ is constructed from the orthonormal basis or via representation theorems. For Clifford- or bicomplex-valued settings, analogous kernels and projection operators exist and are constructed via the inner product and the symmetry properties defined by the underlying algebra (Campos et al., 2018, Vicente-Benítez, 2024, Delgado, 2023, Vicente-Benítez, 2023).

For Cauchy-type and boundary problems, explicit Fredholm integral equations and Riemann–Hilbert boundary formulations are available even in the presence of singular coefficients (Carleman–Vekua) (Tungatarov, 2014).

5. Role in Physical and Inverse Problems

Vekua equations underpin a variety of physically significant scenarios:

• Schrödinger and conductivity equations: The "main" Vekua equation $$\partial_{\bar z}W = (\partial_{\bar z}f/f)\overline{W}$$ connects solutions to pairs of Schrödinger-type PDEs, and in Clifford/quaternionic contexts, scalar parts correspond to solutions of divergence form elliptic equations (Calderón's inverse problem) (Campos et al., 2012, Delgado, 24 Jan 2026, Delgado, 2023).
• Supersymmetric quantum mechanics: Factorization of the superhamiltonian is realized in terms of Vekua and Bers derivative operators, with formal power (pseudoanalytic) expansions providing completeness and Darboux transformation machinery (Bilodeau et al., 2013).
• Three-dimensional (quaternionic/Clifford) models: Dirac-type and Vekua operators naturally encode Maxwell, Beltrami, and div–curl system representations, with associated spectral kernels and direct inversion formulas (Delgado et al., 2018, Delgado et al., 2023, Delgado, 24 Jan 2026).
• Spectral theory and transmutation: Vekua equations are central to transmutation operator theory, completeness of solution sets for Dirac and Helmholtz equations, and provide constructive approaches to eigenfunction expansions (Campos et al., 2011, Kravchenko et al., 2012).

6. Extensions: Parameter-Dependent, Nonlinear, and Singular Vekua Systems

Parameter-dependent Vekua equations leverage structure polynomials (e.g., $i^2 = -\beta(x,y)i - \alpha(x,y)$) to generalize the underlying algebra, thereby unifying elliptic systems with Vekua representations beyond the scope of constant-coefficient or classical complex structures (Alayón-Solarz, 2011). Nonlinear generalizations are developed via "structural transformations" and K-transformations $w\mapsto w K(z)$, encoding all coefficients into a single structural function and yielding associated nonlinear Laplacians (Wang, 2018). Singular coefficients and explicit treatment of zeros/poles are the focus of the Carleman–Vekua class, with unconditional solvability established in weighted function spaces and precise algebraic count of singularities (Tungatarov, 2014).

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Table: Key Operator Forms and Solution Spaces

Setting	Operator/Form	Typical Solution Space
Complex Plane	$\partial_{\bar z} W = aW + b\overline{W}$	$C^1$ or $L^2$ (Bergman-type)
Clifford/Quaternionic Algebra	$D w = \alpha C[w] + w\beta$	$L^p(\Omega, \mathrm{Cl})$, Hodge theory
Compact Lie Groups/Torus	$P u = L u - A u - B \bar u$	$C^\infty$ (global hypoellipticity)
Bicomplex/Biquaternionic	$\partial_{\bar z} W = a W + b\widetilde{W}$	$L^p$, $A^2$, Hardy, or formal-power

This structure demonstrates the unifying character of the Vekua equation as a framework for advancing both abstract PDE theory and concrete analytic techniques in fields such as spectral theory, electrical impedance tomography, quantum mechanics, and harmonic analysis. For explicit characterizations, solvability criteria, and kernel constructions, see (Kirilov et al., 2023, Campos et al., 2018, Delgado, 24 Jan 2026, Delgado, 2023, Vicente-Benítez, 2024, Vicente-Benítez, 2023, Blair, 7 May 2025), and related citations.