Mapping properties of Fourier transforms in function spaces, some recent results

We study continuous and compact mappings generated by the Fourier transform

between distinguished Besov spaces Bsp,p(Rn), 1 ≤ p ≤ ∞, and between Sobolev

spaces Hsp (Rn), 1 < p < ∞. Here we rely mainly 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 relation-

ship to weighted spaces. We can also characterise the situation when the Fourier

transform acts as a nuclear operator.

This is joint work with Leszek Skrzypczak (Poznań) and Hans Triebel (Jena).