Vol. 47, No. 2, 1973

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A Hausdorff-Young theorem for rearrangement-invariant spaces

Colin Bennett

Vol. 47 (1973), No. 2, 311–328

The classical Hausdorff-Young theorem is extended to the setting of rearrangement-invariant spaces. More precisely, if 1 p 2,p1 + q1 = 1, and if X is a rearrangement-invariant space on the circle T with indices equal to p1, it is shown that there is a rearrangement-invariant space X on the integers Z with indices equal to q1 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 q1,2 < q , there is a rearrangement-invariant space Y on T with indices equal to p1 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.

Mathematical Subject Classification 2000
Primary: 42A18
Secondary: 43A15
Received: 5 April 1972
Published: 1 August 1973
Colin Bennett