  We prove mixed $L_{p}(L_{q})$-estimates, with $p,q\in(1,\infty)$, for
higher-order elliptic and parabolic equations on the half space $\R^{d+1}_{+}$
with general boundary conditions which satisfy the Lopatinskii--Shapiro
condition. We assume that the elliptic operators $A$ have leading coefficients
which are in the class of vanishing mean oscillations in both the time and the
space variable. In the proof, we produce mean oscillation estimates for
equations on the half space with general boundary conditions.
