A Characterization of Multipliers of the Herz Algebra

Abstract

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