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