Embeddability of ${\mathrm{\ell }}_p$-spaces into mixed-norm Lebesgue spaces in connection with the validity of vector-valued extensions of the Riesz--Fischer Theorem

The aim of this paper is twofold. On the one hand, we compute, in terms of $r$ and $s$, the indices $p$ for which ${\mathrm{\ell }}_p$ isomorphically embeds into the mixed-norm separable spaces $L_s(L_r)$, ${\mathrm{\ell }}_s(L_r)$, $L_s({\mathrm{\ell }}_r)$ and ${\mathrm{\ell }}_s({\mathrm{\ell }}_r)$. 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 ${\mathrm{\ell }}_2(L_r)$, $r≠2$.