Fourier analysis originated two centuries ago while trying to understand which periodic functions can be decomposed into a sum of undulatory functions: sines and cosines. The attempt to find the precise necessary and sufficient conditions which guaranteed such a representation was a true motor of developments in mathematics in the nineteenth century; for example, both the Riemann and Lebesgue integration theories and the Cantor set theory originated while trying to understand this representation better. Furthermore, it continues to be used frequently as a tool in science and engineering, from signal transmissions to quantum mechanics.
This field continues to be very active and now also considers nonperiodic functions, in which case the sums of undulatory functions are replaced by integrals. We are particularly interested in understanding when we can restrict meaningfully to surfaces such as the cone or the sphere. This subfield of Fourier analysis, called Fourier restriction theory, is of fundamental importance. Many mathematicians, including three Fields Medal awardees, have made recent contributions.
A key new tool in Fourier restriction theory are the recently discovered multilinear estimates. The main objective of the project was to further develop the multilinear approach of Fourier restriction theory. Specifically, the project aims to develop the multilinear restriction estimates with sharp dependence on the transversality and apply such estimates to the Schrödinger and wave equations, to inverse problems, as well as to the linear Fourier restriction problem.
Despite the project being shortened from two years to ten months (lasting from 01/08/2019 to 31/08/2020 with three months paternity leave), we were able to further understanding of the multilinear estimates in three dimensions by proving a refinement, which will likely be helpful to solve –or at least partially solve– the previously mentioned problems, as well as others.
