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.