Let G be a compact abelian group with dual group Γ. For 1 ≤ p <∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) [formula omitted] (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.
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