Some nonlinear Brascamp–Lieb inequalities and applications to harmonic analysis

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.