On the conjugacy of the space of multipliers

Abstract

A. Figà Talamanca proved (1965) that the space Mr = Mr(G) of bounded linear operators in the space Lr, 1 ≤ r ≤ ∞, on a locally compact group G that are translation invariant (more exactly, invariant under the group operation) is the conjugate space for a space Ar = Ar(G), which he described constructively. In the present paper, for the space Mr = Mr(Rm) of multipliers of the Lebesgue space Lr(Rm), 1 ≤ r < ∞, we present a Banach function space Fr = Fr(Rm) with two properties. The space Mr is conjugate to Fr: Fr ∗ = Mr; actually, it is proved that Fr coincides with Ar = Ar(Rm). The space Fr is described in different terms as compared to Ar. This space appeared and has been used by the author since 1975 in the studies of Stechkin's problem on the best approximation of differentiation operators by bounded linear operators in the spaces Lγ(Rm), 1 ≤ γ ≤ ∞. © 2019 Krasovskii Institute of Mathematics and Mechanics. All right reserved.