We have a canonical isomorphism $$ C^0(X,C^0(X,Y)) \simeq C^0(X \times Y, Z)$$ given by $f \mapsto \hat{f}$, where $\hat{f}(x,y) = (f(x))(y)$.
Is there a similar statement for Sobolev space? For example is it true that $$H^1(X, H^1(Y,Z) ) \simeq H^1(X \times Y, Z)$$
(notations : $H^1 = W^{1,2} = L^2_1$)
Your example isn't true: in the space on the left-hand side, you have mixed derivatives of second order (e.g. w.r.t. $X$ and $Y$). You would rather have $$ H^1(X,L^2(Y,Z)) \cap L^2(X, H^1(Y,Z)) = H^1(X \times Y, Z). $$ If I do not miss something, this should just follow from the definitions.