Saturation on locally compact abelian groups: An extended theorem

Abstract

Let G be a locally compact abelian group. The concern of the present note is to extend (for exponents p>2) the saturation theorem on G stated as Theorem 4 in [5]. The extension will be established for approximation processes (It)t>0 acting on the submodule CP(G), p∈]1,+∞[, of the convolutionM1(G)-module LP(G) which consists of all functions f∈LP(G) admitting as their Fourier transformsFGf (in the sense of the theory of quasimeasures) complex Radon measures not necessarily absolutely continuous with respect to any Haar measure on the dual group Ĝ. Moreover, the relationship of the complex vector spaces CP(G) to some other function spaces, in particular to the vector spaces BP(G) introduced in [5], will be investigated.