Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fourier Analysis on $\mathbb{T}^m\times\mathbb{R}^n$ and Applications to Global Hypoellipticity

André Pedroso Kowacs

TL;DR

This work develops a mixed Fourier analysis framework on the product space $\mathbb{T}^m\times\mathbb{R}^n$ by introducing partial Fourier transforms in each factor and a corresponding Schwartz space $\mathcal{S}(\mathbb{T}^m\times\mathbb{R}^n)$. It provides a precise decay-based characterization of $\mathcal{S}(\mathbb{T}^m\times\mathbb{R}^n)$ via bounds on partial Fourier coefficients and extends the theory to tempered distributions, enabling a robust analysis of global regularity for differential operators on $\mathbb{T}^m\times\mathbb{R}^n$. The paper then gives necessary and sufficient conditions for Schwartz global hypoellipticity (SGH) of first-order constant-coefficient operators on $\mathbb{T}^m\times\mathbb{R}^n$, via the zero-set criterion $Z=\{(k,\xi): p(\xi)+q(k)=0\}$, and demonstrates how variable coefficients in the real case can be reduced to the constant-coefficient model by conjugation with automorphisms that preserve the Schwartz space. These results establish a concrete, transferable approach for global regularity analysis on product manifolds and pave the way for generalizations to higher-dimensional torus and Euclidean factors $m,n$.

Abstract

This article presents a convenient approach to Fourier analysis for the investigation of functions and distributions defined in $\mathbb{T}^m \times \mathbb{R}^n$. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus for the initial variables and partial Fourier transform in Euclidean space for the remaining variables. By examining the behaviour of the mixed Fourier coefficients, we achieve a comprehensive characterization of the spaces of fast decaying smooth functions and distributions in this context. Additionally, we apply our results to derive necessary and sufficient conditions for the Schwartz global hypoellipticity of a class of differential operators defined on $\mathbb{T} \times \mathbb{R}$, including all constant coefficient first order differential operators.

Fourier Analysis on $\mathbb{T}^m\times\mathbb{R}^n$ and Applications to Global Hypoellipticity

TL;DR

This work develops a mixed Fourier analysis framework on the product space by introducing partial Fourier transforms in each factor and a corresponding Schwartz space . It provides a precise decay-based characterization of via bounds on partial Fourier coefficients and extends the theory to tempered distributions, enabling a robust analysis of global regularity for differential operators on . The paper then gives necessary and sufficient conditions for Schwartz global hypoellipticity (SGH) of first-order constant-coefficient operators on , via the zero-set criterion , and demonstrates how variable coefficients in the real case can be reduced to the constant-coefficient model by conjugation with automorphisms that preserve the Schwartz space. These results establish a concrete, transferable approach for global regularity analysis on product manifolds and pave the way for generalizations to higher-dimensional torus and Euclidean factors .

Abstract

This article presents a convenient approach to Fourier analysis for the investigation of functions and distributions defined in . Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus for the initial variables and partial Fourier transform in Euclidean space for the remaining variables. By examining the behaviour of the mixed Fourier coefficients, we achieve a comprehensive characterization of the spaces of fast decaying smooth functions and distributions in this context. Additionally, we apply our results to derive necessary and sufficient conditions for the Schwartz global hypoellipticity of a class of differential operators defined on , including all constant coefficient first order differential operators.
Paper Structure (5 sections, 17 theorems, 100 equations)

This paper contains 5 sections, 17 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Partial Fourier Analysis
  3. Application on regularity of Differential Operators
  4. Constant Coefficient case
  5. Variable Coefficients - Real case

Key Result

Proposition 2.1

A function $f\in C^\infty(\mathop{\mathrm{\mathnormal{\mathbb{T}^{m}\times\mathbb{R}^n}}}\nolimits)\cap L^1(\mathop{\mathrm{\mathbb{T}}}\nolimits^m\times\mathop{\mathrm{\mathbb{R}}}\nolimits^n)$ is in $\mathcal{S}(\mathop{\mathrm{\mathnormal{\mathbb{T}^{m}\times\mathbb{R}^n}}}\nolimits)$ if and only for every $k\in\mathop{\mathrm{\mathbb{Z}}}\nolimits^m,$$x\in\mathop{\mathrm{\mathbb{R}}}\nolimits^

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Remark 2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.1
  • Proposition 2.3
  • ...and 29 more