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

TL;DR

This work sharpens the understanding of the non-compactness of the Fourier transform between Banach spaces by proving that for $1<p<2$, the map $\mathscr F: L^p(\mathbb{R}^d) \to L^{p',p}(\mathbb{R}^d)$ is not strictly singular, confirming the optimality of both source and Lorentz target spaces. It employs a construction of an infinite-dimensional subspace generated by carefully scaled test functions whose Fourier images concentrate on disjoint frequency blocks, ensuring a uniform lower bound on $\|\mathscr F f\|_{p',p}$ while keeping $\|f\|_p$ controlled. A parallel discrete version is established for the torus, showing the discrete FT is not strictly singular into $\ell^{p',p}(\mathbb{Z}^d)$, while reinforcing the broader picture that Bernstein-number based finitary non-compactness results do not capture the full non-compactness in the Lorentz-scale target. The paper also discusses extensions to $L^{p,q}$ targets and raises open questions about intermediate Lorentz scales and rearrangement-invariant spaces.

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~$\mathcal{F}$.

Theorems & Definitions (18)

• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 3.1
• Remark 3.2
• Theorem 3.3