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
TL;DR
This work addresses when ell_p embeds into mixed-norm Lebesgue spaces to understand vector-valued extensions of classical theorems. It introduces and analyzes the index set $\bm{\Lambda}(\mathbb{X})$ of embeddings, develops embedding techniques using disjointly supported sequences, the gliding-hump method, and small perturbations, and derives detailed, case-by-case characterizations for mixed-norm spaces such as $L_s(L_r)$, $\ell_s(L_r)$, $L_s(\ell_r)$, $\ell_s(\ell_r)$, and related spaces. A central result is a criterion showing $L_2(\mathbb{X})$ and $\ell_2(\mathbb{X})$ are not isomorphic under broad conditions, contributing to the understanding of when vector-valued Riesz–Fischer-type theorems fail. The paper then uses these $\bm{\Lambda}$-descriptions to advance the isomorphic classification of mixed-norm Lebesgue spaces, producing nearly complete dichotomies for many families and outlining open problems for remaining cases. Overall, the results illuminate the geometry of mixed-norm spaces and provide a framework to distinguish isomorphism classes via $\ell_p$-embeddability.
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 $\ell_p$ isomorphically embeds into the mixed-norm separable spaces $L_s(L_r)$, $\ell_s(L_r)$, $L_s(\ell_r)$ and $\ell_s(\ell_r)$. 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 $\ell_2(L_r)$, $r\not=2$.