Mutually non isomorphic mixed-norm Lebesgue spaces

TL;DR

This paper establishes that mixed-norm Lebesgue spaces are generally non-isomorphic, except when the inner space is $L_2$, highlighting the distinct structure of these function spaces.

Contribution

It proves the non-isomorphism of mixed-norm Lebesgue spaces $L_q(L_p)$ for most parameters, except for a specific isomorphism case involving $L_2$.

Findings

$L_q(L_p)$ spaces are mutually non-isomorphic for most $p,q$

The only exception is $L_q(L_2)$ is isomorphic to $L_q(L_q)$ for all $1<q<inite$

Clarifies the structure and relationships among mixed-norm Lebesgue spaces

Abstract

We prove that for $1\le p,q\le\infty$ the mixed-norm spaces $L_q(L_p)$ are mutually non-isomorphic, with the only exception that $L_q(L_2)$ is isomorphic to $L_q(L_q)$ for all $1<q<\infty$.