Extension of multipliers by periodicity

A theorem proved by de Leeuw for Γ = Rn and later generalized by Lohoué and Saeki states that if Γ is an LCA group, Γ0 a closed subgroup thereof, π the canonical mapping from Γ onto Γ/Γ0 and Φ a Fourier multiplier of type (p, p) on Γ/Γ0, then Φ ० π is a Fourier multiplier of type (p, p) on Γ. We show here that if 1 ≤ p < q ≤ ∞, Γ and Φ is a Fourier multiplier of type (p, q) on Γ/Γ, then Φ ० π is a Fourier multiplier of type (p, q) on Γ and if Γ0 is a non-compact subgroup of Γ and Φ ० π is a Fourier multiplier of type (p, q) on Γ for some p and q satisfying 1 ≤ p ≤ q ≤ ∞, then Φ is zero. We prove also that if Φ is a Fourier multiplier of type (p, q) and Γ/Γ0, whee 1 ≤ q < p ≤ ∞ and Γ is discrete, then Φ ० π is a Fourier multiplier of type (p, q) on Γ.