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

We establish that the Fourier transform $\mathcal{F}:L^p({\mathbb{R}}^d)\to L^{p^{\mathrm{\prime }},p}({\mathbb{R}}^d)$, 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 $\mathcal{F}:L^p({\mathbb{R}}^d)\to L^{p^{\mathrm{\prime }}}({\mathbb{R}}^d)$ is finitely strictly singular and then also strictly singular, and provide further insight into the degrees of non-compactness of~$\mathcal{F}$.