  The purpose of this paper is to establish an atomic decomposition for
functions in the weighted mixed norm space $A^{p,q}_\omega$ induced by a radial
weight $\omega$ in the unit disc admitting a two-sided doubling condition. The
obtained decomposition is further applied to characterize Carleson measures for
$A^{p,q}_\omega$, and bounded differentiation operators $D^{(n)}(f)=f^{(n)}$
acting from $A^{p,q}_\omega$ to $L^p_\mu$, induced by a positive Borel measure
$\mu$, on the full range of parameters $0<p,q,s<\infty$.
