Diogo Oliveira e Silva

On trilinear oscillatory integrals

We examine a certain class of trilinear integral operators which incorporate oscillatory factors e^{iP}, where P is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations.

A sharp nonlinear Hausdorff-Young inequality for small potentials

The nonlinear Hausdorff-Young inequality follows from the work of Christ and Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently of the exponent. We show that the nonlinear Hausdorff-Young quotient admits an even better upper bound than the linear one, provided that the function is sufficiently small in the L^1-norm. The proof combines perturbative techniques with the sharpened version of the linear Hausdorff-Young inequality due to Christ.