Estimates for a family of multi-linear forms

We consider a special class of the multi-linear forms studied by Brascamp and Lieb (1976) [7]. For these forms, we give a characterization of the Lp spaces for which the form is bounded. This characterization appears in earlier work of Barthe (1998) [1]. We use this characterization to study a non-linear map that arises in scattering theory.