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

TL;DR

This paper investigates when classical sequence spaces embed into mixed-norm Lebesgue spaces, and uses these results to classify such spaces, especially distinguishing between certain $L_2$ and $oldsymbol{ ext{ell}_2}$ spaces.

Contribution

It provides a detailed characterization of embeddability of $oldsymbol{ ext{ell}_p}$ into mixed-norm spaces and advances the classification of these spaces based on embedding properties.

Findings

Identifies the values of p for which $oldsymbol{ ext{ell}_p}$ embeds into various mixed-norm spaces.

Distinguishes between spaces $L_2(L_r)$ and $oldsymbol{ ext{ell}_2(L_r)}$ for $r eq 2$.

Contributes to the isomorphic classification of mixed-norm Lebesgue spaces.

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 $\ell_2(L_r)$, $r\not=2$.