Let $(p,q)$ be an admissible pair if $2/p+n/q=n/2$, where $n$ is the space dimension and $2\leq q\leq \frac{2n}{n-2}$. Define $S^0=\{ u, sup_{(p,q)~admissible}||u||_{L^pL^q} <\infty\} $.
I want to show the norm on the dual of $S^0$ is equivalent to $inf_{(p,q)~admissible} ||v||_{L^{p'}L^{q'}}$, where $p'$ and $q'$ are the conjugate indexes of $p$ and $q$.
I'm not sure, but I think the hipothesys on $(p,q)$ are not relevant. This equivalence seems quite common, to be fair.
I looked for references but they usually just give the equivalence.
One can easily prove that the finite inf norm implies being on the dual, but the opposite implication I can't solve.
I'd appreciate any suggestions.