Mixed Lebesgue spaces: information needed Let $\Omega_t$ and $\Omega_x$ be two $\sigma$-finite measure spaces. If it makes things easier we can assume that $\Omega_t$ is some interval and $\Omega_x$ some Euclidean space. For each measurable $f\colon \Omega_t \times \Omega_x \to \mathbb{C}$ define 
$$\lVert f(t, x) \rVert_{L^p_t L^q_x}=\left[\int_{\Omega_t}dt \left(\int_{\Omega_x} \lvert f(t, x) \rvert^q\, dx\right)^\frac{p}{q}\right]^{\frac{1}{p}}.$$
I would like to get some information on the resulting space $L^p_t L^q_x(\Omega_t \times \Omega_x)$, especially: 


*

*Under what name is it known?

*Is it the product of some canonical construction? Is there some obvious way to show that it is complete (if true)?

*Is there any relationship between $\lVert f(t, x) \rVert_{L^p_t L^q_x}$ and $\lVert f(t, x) \rVert_{L^q_x L^p_t}$? Under what circumstances do they coincide?


I'm especially after some reference, but answers of any other kind (proofs, hints, conjectures) are welcome. Thank you.
 A: For point 1 (nomenclature): sometimes you see them called anisotropic Lebesgue spaces, and sometimes in the partial differential equations literature you see them called Strichartz spaces. Most of the time no names are given, and in the PDE context they are almost exclusively studied on $\Omega_t = \mathbb{R}^n$ and $\Omega_x = \mathbb{R}^m$ (or subsets thereof) with Lebesgue measure. A while back I did some literature search, and unfortunately didn't find much myself outside the PDE literature on these spaces.
For point 2 (completeness): maybe through Banach-space valued functions? I haven't actually thought too hard about this. 
For point 3 (reversing order of integration): they generally don't coincide (using Minkowski's inequality you can show that one embeds into the other when $p\neq q$, and a counterexample for the reversed inequality will extend to a counterexample of the reverse embedding), except when $p = q$, which follows from Fubini. Note that in certain cases (for example the $\ell^p$ norm on finite dimensional vector spaces interpreted as an atomic measure on $n$ points) the global equivalence of $L^p$ and $L^q$ as norms would imply the coincidence of the spaces, so any characterisation needs to have some additional hypotheses.
A: The spaces $L^p_tL^q_x$ are special cases of the more general Bochner spaces $L^p(\Omega;X)$ which are Banach spaces in general and whose dual space is given by $L^{p^\prime}(\Omega;X^\prime)$ for $1<p<\infty$ whenever $X$ has the Radon-Nikodym property, e.g. if $X$ is reflexive.
A very good treatment of these spaces and the Bochner integral is given for instance in Vector-valued Laplace Transforms and Cauchy Problems by Arendt, Batty, Hieber, Neubrander.
As far as I know, one can also interprete $L^p(\Omega;X)$ as (projective) tensor product of $L^p(\Omega)$ and $X$.
Edit: The last paragraph only holds for $p=1$, see the comments below.
