Inclusions among multipliers from Lpr to lq

Let G be a compact abelian group with dual Γ. For 1≤p<∞, define Lpr(G)={f;f∈Lp(G),fˆ∈lr(Γ)}. The space (Lpr(G),lq(Γ)) of multipliers from Lpr(G) to lq(Γ) is the set of functions ϕ on Γ such that ϕfˆ∈lq(Γ) for all f in Lpr(G). Suppose that G is a compact abelian group not totally disconnected and r>2, 1≤p<r′. Then (Lpr(G),lq(Γ))⊂(Lr′r(G),lq(Γ)) where 1≤q<r<∞.