Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s\_x(H^s\_y)$, where the mixed Sobolev norm is defined by
$$ \Vert f\Vert\_{H^s\_x(H^s\_y)} = \Vert \Vert f\Vert\_{H^s\_y}\Vert\_{H^s\_x}.$$ 
In other words, $H^s\_x$ functions taking values in $H^s\_y$. My question is, does $f \in H^s(\mathbb{R}^2)$ automatically imply $f \in H^s\_x(H^s\_y)$? 


It is obvious that $L^2\_x(L^2\_y) = L^2(\mathbb{R}^2)$, but in the case of Sobolev spaces, it is not so clear. Understanding this (relatively simple ?) case will help me understand the general theory better.


Edit: From Lions and Magenes, and also from Michael Renardy's answer below, I think that the correct thing to try to infer is $f \in H^s(\mathbb{R}^2) \Rightarrow f \in H^s\_x(L^2\_y)$. But it is not clear to me how to prove this. 



Mixed (anisotropic) Sobolev spaces
The answer to your question is no. If a function $f$ is in $H^1(R^2)$, it means that $f\_x$ and $f\_y$ are in $L^2$, but if $f$ is in $H^1(H^1)$, it means that $f\_{xy}$ is in $L^2$.


