The classical Hausdorff-Young
theorem is extended to the setting of rearrangement-invariant spaces. More precisely,
if 1 ≦ p ≦ 2,p−1+ q−1= 1, and if X is a rearrangement-invariant space on the circle
T with indices equal to p−1, it is shown that there is a rearrangement-invariant
space X on the integers Z with indices equal to q−1 such that the Fourier
transform is a bounded linear operator from X into X. Conversely, for any
rearrangement-invariant space Y on Z with indices equal to q−1,2 < q ≦∞, there is
a rearrangement-invariant space Y on T with indices equal to p−1 such that 𝒯 is
bounded from Y into Y.
Analogous results for other groups are indicated and examples are discussed when
X is Lp or a Lorentz space Lpr.
