Quantitative Non-Compactness Properties of the Fourier Transform on Optimal Spaces

TL;DR

This paper investigates the non-compactness properties of the Fourier transform on optimal Lorentz spaces, establishing that it is not strictly singular, which confirms the optimality of these spaces for the transform.

Contribution

It proves the Fourier transform is not strictly singular on certain Lorentz spaces, extending understanding of its non-compactness and optimality in these function spaces.

Findings

Fourier transform on $L^p$ to Lorentz spaces is not strictly singular.

Results confirm the optimality of source and target spaces for the Fourier transform.

Provides new insights into the degrees of non-compactness of the Fourier transform.

Abstract

We establish that the Fourier transform $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p',p}(\mathbb{R}^d)$, for $d\in\mathbb{N}$ and $1<p<2$, is not strictly singular, thereby confirming the optimality of the source and target spaces. A~similar result is obtained for Fourier series on $L^p(\mathbb{T}^n)$, with sequence Lorentz spaces as the target. These findings complement known results, which state that $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p'}(\mathbb{R}^d)$ is finitely strictly singular and then also strictly singular, and provide further insight into the degrees of non-compactness of~$\mathcal{F}$.