Let $\Omega \subset \mathbb{R}^n$ be a domain. There are two different types of Sobolev spaces on $\Omega \times \mathbb{R}$ that are used in PDE Theory: one is the Bochner-Sobolev space $W^{1,p}(\mathbb{R}, W^{1,q}(\Omega))$ of Banach-space valued functions; the other is the usual Sobolev space $W^{1,p}(\Omega \times \mathbb{R})$ on the product space.

It seems to me that there should be some kind of simple relation between these two kinds of spaces coming from Fubini-Tonelli. However, none of the standard references that I've looked at seem to state any kind of result along these lines, which makes me somewhat concerned. Can someone give a reference for a theorem that relates these two?

Background: I'm reading a PDE paper which states all of its results in terms of Bochner-Sobolev spaces and I'd like to understand how these results translate back into ordinary Sobolev spaces.

  • $\begingroup$ Is there some way I can move this to MO? $\endgroup$ – Maxim Aug 28 '16 at 0:47

The space $W^{1,p}(\mathbb{R};W^{1,q}(\Omega))$ involves derivatives up to order $2$, so this is not the right space to compare with $W^{1,p}(\mathbb{R}\times \Omega)$. But you have that an equivalent norm on $W^{1,p}(\mathbb{R}\times \Omega)$ is given by $\|\cdot\|_{W^{1,p}(\mathbb{R};L^p(\Omega))} + \|\cdot\|_{L^p(\mathbb{R};W^{1,p}(\Omega))}$. The first norm gives information on the time derivative, the latter of the spatial derivatives. If $\Omega$ is bounded, you can get embedings involving an-isotropic spaces as in you question (i.e. with $q$ instead of $p$ for suitable $q$).


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