A Characterization of Multipliers of the Herz Algebra

For the characterization of multipliers Lp(Rd) or more generally, Lp(G) for some locally compact Abelian group G, so-called Figa-Talamanca–Herz algebra Ap(G) plays an important role. Following Larsen’s book, we describe as bounded linear operators that commute with translations. The main result this paper is Ap(G). In fact, demonstrate it coincides space Lp(G),?·?p. Given a multiplier T (Ap(G),?·?Ap(G)) and using embedding (Ap(G),?·?Ap(G))?C0(G),?·??, functional f?[T(f)(0)] bounded, can be written moving average element in dual PMp(G) (Ap(G),?·?Ap(G)). A key step identification another elementary fact: showing from Lp(G),?·?p to C0(G),?·?? are exactly convolution kernels Lq(G),?·?q 1<p<? 1/p+1/q=1. proofs make use mild distributions, which Segal S0(G),?·?S0, fact S0(G) S0?(G) form T:f???f uniquely determined ??S0?. This setting also allows us switch description these (by suitable pseudomeasures) their Fourier multipliers, extended transform S0?(G),?·?S0?. approach presented here extends other function spaces, but detailed discussion left future publications.