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A Fourier analysis for $(θ,T)$-periodic functions and applications

André Pedroso Kowacs, Marielle Aparecida Silva

TL;DR

This work generalizes Fourier analysis to the class of $(\theta,T)$-periodic functions, introducing the transform $\Omega_{\theta,T}$ and building corresponding $L^p$, Sobolev, and distribution frameworks. It establishes a Poincaré-type inequality in this setting and shows that global regularity properties for continuous linear operators on $(\theta,T)$-periodic functions can be fully characterized by the associated torus problem. The key mechanism is the reduction to the classical periodic case via the induced operator $\tilde{P}$ on $\mathbb{T}^n$, enabling explicit criteria for global hypoellipticity and solvability, including for first-order differential operators with periodic coefficients. This unifies and extends existing periodic and Floquet-type analyses, with potential applications to differential equations exhibiting phase symmetry.

Abstract

We develop a Fourier analysis for a generalization of the class of periodic functions, often referred to as $(θ, T)$-periodic functions, and prove several properties and inequalities related to the Fourier transform, including a type of Poincaré inequality, which extend the periodic case. As an application, we employ this analysis to show that a continuous linear operator acting on smooth $(θ, T)$-periodic functions is globally hypoelliptic/solvable if and only if the corresponding operator which acts on periodic functions is globally hypoelliptic/solvable, and characterize the global hypoellipticity/solvability of a class of first order differential operators acting on the set of smooth $(θ, T)$-periodic functions.

A Fourier analysis for $(θ,T)$-periodic functions and applications

TL;DR

This work generalizes Fourier analysis to the class of -periodic functions, introducing the transform and building corresponding , Sobolev, and distribution frameworks. It establishes a Poincaré-type inequality in this setting and shows that global regularity properties for continuous linear operators on -periodic functions can be fully characterized by the associated torus problem. The key mechanism is the reduction to the classical periodic case via the induced operator on , enabling explicit criteria for global hypoellipticity and solvability, including for first-order differential operators with periodic coefficients. This unifies and extends existing periodic and Floquet-type analyses, with potential applications to differential equations exhibiting phase symmetry.

Abstract

We develop a Fourier analysis for a generalization of the class of periodic functions, often referred to as -periodic functions, and prove several properties and inequalities related to the Fourier transform, including a type of Poincaré inequality, which extend the periodic case. As an application, we employ this analysis to show that a continuous linear operator acting on smooth -periodic functions is globally hypoelliptic/solvable if and only if the corresponding operator which acts on periodic functions is globally hypoelliptic/solvable, and characterize the global hypoellipticity/solvability of a class of first order differential operators acting on the set of smooth -periodic functions.

Paper Structure

This paper contains 8 sections, 19 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Analysis on (θ,T)-periodic functions
  3. A Fourier transform for (θ,T)-periodic functions
  4. (θ,T)-periodic Sobolev spaces
  5. Poincaré Inequality
  6. Global regularity of continuous linear operators
  7. Global regularity for a class of first order differential operators on the torus
  8. (θ,T)-periodic solutions to a class of first order differential equations

Key Result

Proposition 2.5

For every $f\in C^\infty_{\theta,T}(\mathbb{R}^n)\subset \mathcal{D}'_{\theta,T}(\mathbb{R}^n)$, we have that that is, Analogously, for every $f\in C^\infty(\mathbb{T}^n)$, we have that that is,

Theorems & Definitions (49)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 1
  • Remark 2
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Remark 3
  • ...and 39 more