Extensions of the Hausdorff-Young theorem

In this paper the clasical Hausdorff-Young theorem, which states that iff ∈L ^p, 1≦p≦2, on the line and is its Fourier transform, then whereq ⁻¹+p ⁻¹=1, is extended in two ways for certain Orlicz spacesL ^Φ. IfL ^Φ is based on (G, μ), (1) an arbitrary compact topological group with Haar measure, and (2) a locally compact abelian topological group andμ is again the Haar measure, then the above inequality is extended to these cases. Various other related results and remarks are also included. Dedicated to the memory of my nephew, K. Ramakrishna, who appeared to be so brilliant. This research was supported by the NSF Grants GP-5921 and GP-7678.