  In this note we prove a multilinear version of the reverse H\"older
inequality in the theory of Muckenhoupt $A_p$ weights. We give two applications
of this inequality to the study of multilinear weighted norm inequalities.
First, we prove a structure theorem for multilinear $A_{\vec{p}}$ weights;
second, we give a new sufficient condition for multilinear, two-weight norm
inequalities for the maximal operator.
