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 $H^s_x(H^s_y)$. My question is, does $f \in H^s(\mathbb{R}^2)$ automatically imply $f \in H^s_x(H^s_y)$?