Denjoy-Carleman solvability of Vekua-type periodic operators

1. Problem Statement

The paper investigates the solvability and global hypoellipticity of Vekua-type differential operators defined on the $n$-dimensional torus $\mathbb{T}^n$. The study is conducted within the framework of Denjoy-Carleman ultradifferentiable classes (specifically of Roumieu type), denoted as $E_M(\mathbb{T}^n)$.

The primary operator under consideration is:
$$Pu = Lu - Au - B\bar{u}$$
where:

• $L$ is a partial differential operator (initially with constant coefficients, later extended to variable coefficients).
• $A$ and $B$ are complex constants.
• $\bar{u}$ denotes the complex conjugate of $u$.

The authors aim to establish necessary and sufficient conditions for the operator $P$ to be $M$-solvable (solvability within the ultradifferentiable class) and $M$-globally hypoelliptic (regularity of solutions within the same class). This extends previous work on smooth ($C^\infty$) solvability and Gevrey solvability to the broader and more general Denjoy-Carleman setting.

2. Methodology

The authors employ a combination of Fourier analysis, symbolic calculus, and Diophantine approximation techniques tailored to ultradifferentiable spaces.

A. Framework of Denjoy-Carleman Classes

The paper utilizes the characterization of functions $f \in E_M(\mathbb{T}^n)$ and ultradistributions $u \in D'_M(\mathbb{T}^n)$ via the decay and growth rates of their Fourier coefficients.

• A function $f$ belongs to $E_M$ if its Fourier coefficients $\hat{f}(\xi)$ decay rapidly, controlled by a weight sequence $M = \{m_j\}$ and a parameter $h$.
• The condition for solvability relies on the behavior of the discriminant $\Delta_\xi$ associated with the operator's symbol.

B. Constant Coefficient Case

For operators with constant coefficients, the authors transform the differential equation $Pu = f$ into an algebraic system in the frequency domain (Fourier space).

1. System Reduction: By taking the Fourier transform, the equation $Pu=f$ leads to a linear system for the coefficients $\hat{u}(\xi)$ and $\hat{u}(-\xi)$.
2. Discriminant Analysis: The solvability depends on the invertibility of this system, governed by the discriminant:
   $$\Delta_\xi = (\sigma_L(\xi) - A)(\sigma_L(-\xi) - \bar{A}) - |B|^2$$
   where $\sigma_L$ is the symbol of $L$.
3. Diophantine Conditions: The authors derive a specific growth condition on $|\Delta_\xi|$ relative to the weight sequence $M$. This condition, labeled (DCM), ensures that the inverse of the discriminant does not grow too fast to destroy the ultradifferentiable regularity of the solution.

C. Variable Coefficient Case

For operators with variable coefficients, the authors consider a complex vector field $L$ satisfying the Nirenberg-Treves condition (P) (ensuring local solvability).

1. Transformation: They utilize a change of variables (an automorphism $T$) to reduce the variable coefficient operator to a form with constant coefficients in the spatial variables but time-dependent coefficients.
2. Partial Fourier Series: The analysis is performed using partial Fourier series with respect to the spatial variables, reducing the problem to a system of ordinary differential equations in the time variable.
3. Explicit Construction: The solution is constructed explicitly using integral representations involving the coefficients of the vector field. The proof involves rigorous estimates using Faà di Bruno's formula to control the derivatives of composite functions, ensuring the solution remains in the Denjoy-Carleman class.

3. Key Contributions and Results

A. Constant Coefficient Operators

• Theorem 3.2 (Main Result): The operator $P$ is $M$-solvable if and only if the discriminant $\Delta_\xi$ satisfies the Diophantine-type condition (DCM):
  $$\|\xi\| \geq \gamma_\varepsilon \implies |\Delta_\xi| \geq C_\varepsilon \inf_{j \in \mathbb{N}_0} \frac{m_j j!}{\varepsilon^j (1+\|\xi\|)^j}$$
  This condition links the arithmetic properties of the operator's symbol to the growth of the weight sequence $M$.
• Equivalence of Properties: The paper proves that for this class of operators, $M$-solvability is equivalent to $M$-global hypoellipticity (Corollary 3.4).
• Comparison with Smooth Solvability: The authors show that smooth solvability implies $M$-solvability (Proposition 3.5), but the converse is not generally true. This highlights that ultradifferentiable solvability is a strictly weaker condition than smooth solvability in certain contexts (e.g., involving Liouville numbers).
• Applications: The results are applied to specific classical operators:
  • Laplace Operator: Always $M$-solvable (due to ellipticity).
  • Heat Operator: Always $M$-solvable.
  • Wave Operator: Solvability depends on specific relations between $A, B$, and the irrationality properties of the wave speed parameter $\eta$.

B. Variable Coefficient Operators

• Theorem 4.1: Establishes solvability for a class of Vekua-type operators with variable coefficients satisfying condition (P). The conditions involve:
  1. Non-degeneracy of parameters ($|\alpha| \neq |\delta|$).
  2. Absence of specific resonant solutions in the Fourier domain.
  3. A Diophantine condition (DCM) on the exponential terms arising from the variable coefficients.
• Theorem 4.2: Provides a detailed classification for the case where the vector field has no spatial rotation ($\lambda = 0$). It identifies four distinct scenarios (based on the magnitude of $A_0, B_0$ and the Diophantine properties of the coefficients) under which the operator is solvable in $C^\infty$ (and by extension, in $E_M$).

4. Significance

• Generalization: This work significantly generalizes previous results on Vekua-type operators. While earlier studies focused on $C^\infty$ or Gevrey classes ($G^s$), this paper addresses the full generality of Denjoy-Carleman classes, which include both Gevrey and non-quasianalytic classes.
• Unified Framework: It provides a unified necessary and sufficient condition (DCM) that characterizes solvability across different types of ultradifferentiable spaces, linking the spectral properties of the operator to the arithmetic nature of the torus.
• Technical Rigor: The paper offers a rigorous treatment of variable coefficient operators in ultradifferentiable spaces, a domain where technical challenges (such as controlling the growth of derivatives of composite functions) are substantial. The explicit use of Faà di Bruno's formula and careful estimation of Fourier coefficients demonstrates the depth of the analysis.
• Applications: The results have implications for mathematical physics, particularly in the study of generalized analytic functions, elasticity, and hydrodynamics, where Vekua-type equations frequently arise.

In summary, the paper successfully bridges the gap between classical solvability theory and ultradifferentiable analysis, providing a comprehensive characterization of when Vekua-type operators on the torus admit solutions within specific regularity classes.