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Differential Complexes in Time-Periodic Gelfand-Shilov Spaces

Fernando de Ávila Silva, Marco Cappiello, Alexandre Kirilov, Pedro Meyer Tokoro

TL;DR

This work characterizes the global solvability and hypoellipticity of a differential complex on $\mathbb{T}^m\times\mathbb{R}^n$ generated by first-order evolution operators with time-periodic Gevrey coefficients. By embedding the problem into time-periodic Gelfand–Shilov spaces and reducing to a normal form with constant coefficient $a_0$, the authors derive a sharp Diophantine-type condition ($\mathrm{DC}_{\sigma,\mu}$) relating the constant part of the 1-form to the spectrum of a globally elliptic operator $P$. They prove that global solvability is equivalent to this DC condition, with a constructive sufficiency proof and a necessity argument based on resonant modes. Global hypoellipticity, in contrast, exhibits a dichotomy: it holds for $p=0$ under finite resonance and the DC condition, but fails for every $p\ge1$ due to obstructions arising from nontrivial differential forms. The results extend scalar and constant-coefficient theories to time-dependent differential complexes within the Gelfand–Shilov framework, highlighting the interplay between spectral data, arithmetic conditions, and ultradifferentiable regularity.

Abstract

We study the global solvability of a class of differential complexes on the product manifold $\mathbb{T}^m \times \mathbb{R}^n$ associated with systems of evolution operators of the form $L_r = \partial_{t_r} + ia_r(t)P(x,D_x), r=1,\ldots,m,$ where the coefficients $a_r$ are real-valued Gevrey functions on the torus and $P(x,D_x)$ is a globally elliptic normal differential operator on $\mathbb{R}^n$. Within the framework of time-periodic Gelfand--Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated $1$-form and the spectrum of $P$. We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients.

Differential Complexes in Time-Periodic Gelfand-Shilov Spaces

TL;DR

This work characterizes the global solvability and hypoellipticity of a differential complex on generated by first-order evolution operators with time-periodic Gevrey coefficients. By embedding the problem into time-periodic Gelfand–Shilov spaces and reducing to a normal form with constant coefficient , the authors derive a sharp Diophantine-type condition () relating the constant part of the 1-form to the spectrum of a globally elliptic operator . They prove that global solvability is equivalent to this DC condition, with a constructive sufficiency proof and a necessity argument based on resonant modes. Global hypoellipticity, in contrast, exhibits a dichotomy: it holds for under finite resonance and the DC condition, but fails for every due to obstructions arising from nontrivial differential forms. The results extend scalar and constant-coefficient theories to time-dependent differential complexes within the Gelfand–Shilov framework, highlighting the interplay between spectral data, arithmetic conditions, and ultradifferentiable regularity.

Abstract

We study the global solvability of a class of differential complexes on the product manifold associated with systems of evolution operators of the form where the coefficients are real-valued Gevrey functions on the torus and is a globally elliptic normal differential operator on . Within the framework of time-periodic Gelfand--Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated -form and the spectrum of . We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients.
Paper Structure (8 sections, 11 theorems, 118 equations)

This paper contains 8 sections, 11 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenfunction Expansions for Time-Periodic Gelfand–Shilov Spaces
  4. Reduction to Normal Form
  5. Global Solvability
  6. Proof of Theorem \ref{['thm_GS']}: Sufficiency
  7. Theorem \ref{['thm_GS']}: Necessity
  8. Global Hypoellipticity

Key Result

Proposition 2.1

Let $\mu \geq 1/2$ and $\sigma \geq 1$. Then a distribution $u \in \mathcal{S}'_{\sigma,\mu}(\mathbb{T}^m \times \mathbb{R}^n)$ belongs to $\mathcal{S}_{\sigma,\mu}(\mathbb{T}^m \times \mathbb{R}^n)$ if and only if it admits a decomposition where and there exist constants $C > 0$ and $\varepsilon > 0$ such that for all $j \in \mathbb{N}$ and $\gamma \in \mathbb{N}_0^m.$

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • proof
  • ...and 14 more