A converse theorem for Borcherds products and the injectivity of the Kudla-Millson theta lift

We prove a converse theorem for the multiplicative Borcherds lift for lattices of square-free level whose associated discriminant group is anisotropic. This can be seen as generalization of Bruinier's results in \cite{Br2}, which provides a converse theorem for lattices of prime level. The surjectivity of the Borcherds lift in our case follows from the injectivity of the Kudla-Millson theta lift. We generalize the corresponding results in \cite{BF1} to the aforementioned lattices and thereby in particular to lattices which are not unimodular and not of type $(p,2)$. Along the way, we compute the contribution of both, the non-Archimedean and Archimedean places of the $L^2$-norm of the Kudla-Millson theta lift. As an application we refine a theorem of Scheithauer on the non-existence of reflective automorphic products.