Rearrangement-Invariant Spaces of Functions on LCA Groups

In this paper the rearrangement-invariant hulls and kernels of the classes {f: f∈Lp} and {f: f∈L1loc} on a noncompact nondiscrete locally compact abelian group are characterized. The spaceL1loc∩S′ of tempered distributions serves as the extended domain of the Fourier transform instead of the classical domainL1+L2. We also introduce and study a new function spaceP0which arises as the rearrangement-invariant hull of some of the classes under consideration.