A multilinear reverse H\"older inequality with applications to multilinear weighted norm inequalities

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.