%0 Journal Article %T Some nonlinear Brascamp-Lieb inequalities and applications to harmonic analysis %A Jonathan Bennett %A Neal Bez %J Mathematics %D 2009 %I arXiv %X We use the method of induction-on-scales to prove certain diffeomorphism invariant nonlinear Brascamp--Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru--Herr--Tataru and Bennett--Carbery--Wright. %U http://arxiv.org/abs/0906.2064v3