V.V. Arestov. On the conjugacy of the space of multipliers ... P. 5-14

A. Fig$\grave{\mathrm{a}}$ Talamanca proved (1965) that the space $M_r=M_r(G)$ of bounded linear operators in the space $L_r$, $1\le r\le\infty$, on a locally compact group $G$ that are translation invariant (more exactly, invariant under the group operation) is the conjugate space for a space $A_r=A_r(G)$, which he described constructively. In the present paper, for the space $M_r=M_r(\mathbb{R}^m)$ of multipliers of the Lebesgue space $L_r(\mathbb {R}^m)$, $1\le r<\infty$, we present a Banach function space $F_r=F_r(\mathbb{R}^m)$ with two properties. The space $M_r$ is conjugate to $F_r$: $F^*_r=M_r$; actually, it is proved that $F_r$ coincides with $A_r=A_r(\mathbb{R}^m)$. The space $F_r$ is described in different terms as compared to $A_r$. 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 $L_\gamma(\mathbb{R}^m)$, $1\le\gamma\le\infty$.

Keywords: predual space for the space of multipliers

Received September 15, 2019

Revised October 14, 2019

Accepted October 18, 2019

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Vitalii Vladimirovich Arestov, Dr. Phys.-Math. Sci., Ural Federal University, Yekaterinburg, 620083 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: vitalii.arestov@urfu.ru

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Cite this article as: V.V.Arestov. On the conjugacy of the space of multipliers, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 5–14.