Nuclear Fourier transforms

TL;DR

This paper investigates the conditions under which the Fourier transform acts as a nuclear operator between Besov and Triebel-Lizorkin type function spaces, extending previous work on compactness to nuclearity.

Contribution

It provides new criteria for the nuclearity of the Fourier transform between these advanced function spaces, expanding the theoretical understanding of Fourier analysis in this context.

Findings

Identifies conditions for nuclearity of Fourier transform between specific function spaces.

Extends previous results from compactness to nuclearity.

Provides a comprehensive framework for analyzing Fourier transform properties in Besov and Triebel-Lizorkin spaces.

Abstract

The paper deals with the problem under which conditions for the parameters $s_1,s_2\in\mathbb{R}$, $1\leq p,q_1,q_2\leq\infty$ the Fourier transform $\mathcal{F}$ is a nuclear mapping from $A^{s_1}_{p,q_1}(\mathbb{R}^n)$ into $A^{s_2}_{p,q_2}(\mathbb{R}^n)$, where $A\in\{B,F\}$ stands for a space of Besov or Triebel-Lizorkin type, and $n\in\mathbb{N}$. It extends the recent paper arXiv:2112.04896 where the compactness of $\mathcal{F}$ acting in the same type of spaces was studied.