  The paper deals with continuous and compact mappings generated by the Fourier
transform between distinguished Besov spaces $B^s_p(\mathbb{R}^n) =
B^s_{p,p}(\mathbb{R}^n)$, $1\le p \le \infty$, and between Sobolev spaces
$H^s_p(\mathbb{R}^n)$, $1<p< \infty$. In contrast to the paper {\em H. Triebel,
Mapping properties of Fourier transforms. Z. Anal. Anwend. 41 (2022),
133--152}, based mainly on embeddings between related weighted spaces, we rely
on wavelet expansions, duality and interpolation of corresponding (unweighted)
spaces, and (appropriately extended) Hausdorff-Young inequalities. The degree
of compactness will be measured in terms of entropy numbers and approximation
numbers, now using the symbiotic relationship to weighted spaces.
