TITLE: Mixed Lebesgue spaces: information needed
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: 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.