The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces B p s (ℝn) = B p,p s (ℝn), 1 ≤ p ≤ ∞, and between Sobolev spaces H p s (ℝn), 1 < p < ∞. In contrast to the paper 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.
