Mapping properties of Fourier transforms, revisited

Abstract

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces $B_p^s({\mathbb{R}}^n)=B_{p,p}^s({\mathbb{R}}^n)$, $1\le p\le \infty$, and between Sobolev spaces $H_p^s({\mathbb{R}}^n)$, $1<p<\infty$. 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.