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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.

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(BTW, if you don't get any useful answers after a while, you may consider asking this on MathOverflow.) – Willie Wong Apr 26 '11 at 17:00
up vote 4 down vote accepted

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.

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1) Strichartz spaces, ok! In fact (as you surely have guessed from what I'm asking those days) I'm studying Strichartz estimates for the Schrödinger equation. [...] – Giuseppe Negro Apr 26 '11 at 17:16
2) I will think about it a little. My guess is that they are Banach spaces and that we have the representation $[L^p_tL^q_x]'\equiv L^{p'}_tL^{q'}_x$. This would justify the terminology $T, T^\star, TT^\star$ that I see in my course notes. [...] – Giuseppe Negro Apr 26 '11 at 17:16
3) Ok. I guess that for PDE use it is most important to take the spacial norm first. By Minkowski's inequality you mean this one, right? – Giuseppe Negro Apr 26 '11 at 17:19
For (2) you should see lvb's answer. There's still some grunt work to be done (showing that $f$ can be interpreted as a measurable function with values in a Banach space). For (3), yes. – Willie Wong Apr 26 '11 at 17:43

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.

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The last paragraph is incorrect. for instance, one can show that the diagonal of $\ell^2 \hat{\otimes}_{\pi} \ell^2$ is a complemented subspace isomorphic to $\ell^1$. However, $\ell^2(\mathbb{N},\ell^2)$ is a Hilbert space. – t.b. Apr 26 '11 at 18:00
Thank you. I wasn't too sure about that either. I'm pretty sure, though, that the last paragraph holds at least for $p=1$. – lvb Apr 26 '11 at 18:07
Yes, that's true (this is not hard to show by using that the step functions are dense and isometric in both $L^1(\Omega,X)$ and $L^1(\Omega) \hat{\otimes}_{\pi} X$). I think Grothendieck was the first to observe this. There are tensor norms compatible with $L^p$ but I can't seem to remember their name. – t.b. Apr 26 '11 at 18:12
Do you recall any reference for $L^p$-compatible tensor product constructions? – lvb Apr 26 '11 at 18:17
I think there is something to that effect in C. Herz, The theory of $p$-spaces (or a similar title. I can't access MathSciNet at the moment). I might misremember, though. Quite probably things are more subtle. There are also things related to the Lindenstrauss-Pelczinsky $\mathcal{L}_p$-spaces, see e.g. in Ryan and references therein (but again, things are more subtle than one might wish for). – t.b. Apr 26 '11 at 18:29

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