On Lp-Multipliers of Mixed-Norm Type

Given a sequence (gn) of Fourier multipliers for Lp(R), 1 < p < ∞, let g ≔ Σ∞−∞gnχn, where χn denotes the characteristic function of the interval [2n, 2n+1] in R. Assuming (gn) ∈ ℓs(M(p)) for some s with 0 < s ≤ ∞, we determine the values of s for which g is, or is not, a multiplier of Lp(R). Our results sharpen a result of Littman et al. who, in 1968, considered the case when s = ∞. The same problem is also considered for multipliers in Lp-spaces defined on a locally compact Vilenkin group