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Systems of differential operators in time-periodic Gelfand-Shilov spaces

Fernando de Ávila Silva, Marco Cappiello, Alexandre Kirilov

Abstract

This paper explores the global properties of time-independent systems of operators in the framework of Gelfand-Shilov spaces. Our main results provide both necessary and sufficient conditions for global solvability and global hypoellipticity, based on analysis of the symbols of operators. We also present a class of time-dependent operators whose solvability and hypoellipticity are linked to the same properties of an associated time-independent system, albeit with a loss of regularity for temporal variables.

Systems of differential operators in time-periodic Gelfand-Shilov spaces

Abstract

This paper explores the global properties of time-independent systems of operators in the framework of Gelfand-Shilov spaces. Our main results provide both necessary and sufficient conditions for global solvability and global hypoellipticity, based on analysis of the symbols of operators. We also present a class of time-dependent operators whose solvability and hypoellipticity are linked to the same properties of an associated time-independent system, albeit with a loss of regularity for temporal variables.
Paper Structure (8 sections, 15 theorems, 151 equations)

This paper contains 8 sections, 15 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. Time-periodic Gelfand-Shilov spaces and eigenfunction expansions
  4. Full Fourier coefficients
  5. Global hypoellipticity
  6. Global solvability
  7. Examples
  8. A class of time dependent coefficients systems

Key Result

Theorem 1.2

The system ${\mathbb{L}}$ defined by general-const-system is $\mathcal{S}_{\sigma,\mu}$-globally hypoelliptic if and only if $\mathcal{N}$ is finite and for all $\varepsilon>0$, there exists $C_\varepsilon>0$ such that for any $(\tau,j)\in\mathbb{Z}^m\times{\mathbb{N}}$ such that $\sigma_{\mathbb{L}}(\tau,j)\neq0.$

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 22 more