Multipliers from Lp to lq

Let G be a compact abelian group with dual Γ. A function φ on Γ is said to be a multiplier from Lp(G) to lq(Γ) (φ∈(Lp(G),lq(Γ))) if φfˆ∈lq(Γ) for every f∈Lp(G).
Suppose that G is a compact abelian group not totally disconnected and 1≤p1<p2≤2, 1≤q<p′1. Then (Lp1(G),lq(Γ))⊂(Lp2(G),lq(Γ)).
Using the Weil's map the author reduces the problem to the case of G=T. The result is then a consequence of the theory of trigonometric series with decreasing coefficients.
For every compact abelian group G, 1<p<2, q<p′, one has lp′q/(p′−q)(Γ)⊆(Lp(G),lq(Γ)).
For G=T, 1<p<2, q<p′, φ∈(Lp(T),lq(Z)) non-negative, decreasing and even, he proves that φ∈lr(Z) for all r>p′q/(p′−q). The author also obtains a sufficient condition on a sequence φ to be in (Lp(T),lq(Z)) (Theorem 8 p. 157). As a consequence he shows the existence of φ∈(Lp(T),lq(Z)) non-negative, decreasing and even but not in lp′q/(p′−q)(Z).