On the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$, or: Interchange of supremum and integral

Question

I am asking whether the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$ are the same. Or, whether one is included/embedded in the other.

We have the norms

$$\|u\|_{L^\infty L^p}=\sup_{t \in (a,b)} \left(\int_c^d |u(t,s)|^pds \right)^{1/p}$$

$$\|u\|_{L^pL^\infty}= \left(\int_c^d \left(\sup_{t \in (a,b)} |u(t,s)|\right)^pds \right)^{1/p}$$

So the question boils down to interchanging (ess)sup and integral. I looked at this question on MSE: https://math.stackexchange.com/questions/2852005/interchange-between-integral-and-sup It says as a sufficient and necessary condition that I need to construct a sequence $t_n$ such that $u(t_n,s) \to \sup_t u(t,s)$ uniformly. I am not sure how to do that.

At least I know that it holds $|u| \leq \sup |u|$, and hence $\int |u| \leq \int \sup |u|$ and hence $\sup \int |u| \leq \int \sup |u|$. This should give me at least $\|u\|_{L^\infty L^p} \leq \|u\|_{L^pL^\infty}$, i.e., $$L^p(c,d;L^\infty(a,b)) \subset L^\infty(a,b;L^p(c,d)).$$ But what about the other direction? Can we prove it or is there a counterexample? I aware that it holds $L^\infty(a,b;L^\infty(c,d)) \subsetneq L^\infty((a,b)\times(c,d))$. Any hint is helpful to me.

Answer 1

If you want a hint, then in such situations it is often advisable to look first at simple, non-trivial cases. Here you can start with a drastically simplified one—I looked at $p=1$ and two point measure spaces where it soon becomes clear what is going on.

Answer 2

Another hint: if you are looking for a more sophisticated argument, then for the case $p=1$ your spaces have well known representations as tensor products, but with different norms (a good reference for this is “Vector measures” by Diestel and Uhl). The same works for $p>1$ but with more esoteric tensor norms which were introduced by Grothendieck.