  Consider the anisotropic Navier-Stokes equations as well as the primitive
equations. It is shown that the horizontal velocity of the solution to the
anisotropic Navier-Stokes equations in a cylindrical domain of height
$\varepsilon $ with initial data $u_0=(v_0,w_0)\in B^{2-2/p}_{q,p}$,
$1/q+1/p\le 1$ if $q\ge 2$ and $4/3q+2/3p\le 1$ if $q\le 2$, converges as
$\varepsilon \to 0$ with convergence rate $\mathcal{O} (\varepsilon )$ to the
horizontal velocity of the solution to the primitive equations with initial
data $v_0$ with respect to the maximal-$L^p$-$L^q$-regularity norm. Since the
difference of the corresponding vertical velocities remains bounded with
respect to that norm, the convergence result yields a rigorous justification of
the hydrostatic approximation in the primitive equations in this setting. It
generalizes in particular a result by Li and Titi for the $L^2$-$L^2$-setting.
The approach presented here does not rely on second order energy estimates but
on maximal $L^p$-$L^q$-estimates for the heat equation.
