Periodic Reporting for period 1 - RESTRICTIONAPP (A multilinear approach to the restriction problem with applications to geometric measure theory, the Schrödinger equation and inverse problems)
Berichtszeitraum: 2019-08-01 bis 2021-07-31
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.
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.
We have obtained a promising result and have some work in progress towards achieving the objectives of the proposal. In particular, we have obtained in three dimensions a refinement of the multilinear restriction estimate for the paraboloid and the cone. In the proposal this was stated as one of the key milestones of the project.
We have also been exploring some decoupling estimates for fractal measures which we expect to be useful for both the problems of Falconer and Carleson.
Unfortunately, the dissemination of the result has been not possible due to the pandemic; my participation in a workshop and visits to several universities have been postponed. Once these events have been rescheduled I will carry out this diffusion.
We have also been exploring some decoupling estimates for fractal measures which we expect to be useful for both the problems of Falconer and Carleson.
Unfortunately, the dissemination of the result has been not possible due to the pandemic; my participation in a workshop and visits to several universities have been postponed. Once these events have been rescheduled I will carry out this diffusion.
We hope to use the obtained refinement of the trilinear restriction estimate in a variety of problems including the ones in the proposal. For the first time, this refinement provides an alternative to the Bourgain-Guth argument, which has been a key tool to solve many problems. In particular we hope to use this refined multilinear theory to prove new linear restriction estimates. As well as improving the state-of-the-art for the restriction conjecture, we also hope to prove some open sharp bilinear estimates. In addition, we expect to prove the refined estimates in higher dimensions.