Rearrangements of Functions on a Locally Compact Abelian Group and Integrability of the Fourier Transform

We find in this paper the equimeasurable hulls and kernels of some function classes on a locally compact abelian group. These classes consist of all functions for which the Fourier transform belongs to a given Lorentz space on the dual group. Different special cases of the problems considered in this paper have been originally studied by Hardy, Littlewood, Hewitt, Ross, Cereteli, and the author.