A. Figà-Talamanca and G. I. Gaudry proved (1967) that the space τp,q (G) of bounded linear operators from the space Lp to the space Lq, 1 ≤ P ≤ q < ∞, on a locally compact Abelian group G that are translation invariant (more exactly, invariant with respect to the group operation) is a conjugate space for a space Ap,q(G), which they described constructively. Figà-Talamanca obtained such result for the case q = p earlier (1965). In this paper, for the space τp,q(ℝm) or, that is the same, for the space of multipliers Mp,q = Mp,q(ℝm)of a pair of Lebesgue spaces Lp(ℝm) and Lq(ℝm), 1 ≤ p ≤ q ≤ ∞,wepresent a Banach functional space Fp,q = Fp,q(ℝm) with two properties: (1) the space Mp,q is conjugate for Fp,q= Fp,q(Rn)(2) the space Fp,q is isometrically isomorphic to the space Ap,q = Ap,q(ℝm). The space Fp,q is described in other terms in comparison with Ap,q. The author obtained in 2019 a similar result for the case q = p. Starting with 1980, the author introduced and used the space Fp,q in the studies of Stechkin’s problem on the best approximation of differentiation operators by bounded linear operators in Lebesgue spaces Lγ(ℝm), 1 ≤ γ ≤ ∞. Subsequently, in a number of the author’s papers, several extremal problems arose and were studied in the spaces Fp,q.