Embeddability of $\ell_p$-spaces into mixed-norm Lebesgue spaces in connection with the validity of vector-valued extensions of the Riesz--Fischer Theorem

José L. Ansorena; Glenier Bello

paper

Embeddability of $ℓ_{p}$ -spaces into mixed-norm Lebesgue spaces in connection with the validity of vector-valued extensions of the Riesz--Fischer Theorem

arXiv:2502.13561

Abstract

The aim of this paper is twofold. On the one hand, we compute, in terms of $r$ and $s$ , the indices $p$ for which $ℓ_{p}$ isomorphically embeds into the mixed-norm separable spaces $L_{s} (L_{r})$ , $ℓ_{s} (L_{r})$ , $L_{s} (ℓ_{r})$ and $ℓ_{s} (ℓ_{r})$ . On the other hand, we use this information to move forward in the isomorphic classification of mixed-norm spaces. In particular, we tell apart the spaces $L_{2} (L_{r})$ and $ℓ_{2} (L_{r})$ , $r \neq = 2$ .

18 pages

Embeddability of ℓp​-spaces into mixed-norm Lebesgue spaces in connection with the validity of vector-valued extensions of the Riesz--Fischer Theorem

Abstract

Embeddability of $ℓ_{p}$ -spaces into mixed-norm Lebesgue spaces in connection with the validity of vector-valued extensions of the Riesz--Fischer Theorem