Fourier multipliers, square functions and incidence theory.

Proposed research proposal*:The goal of this project is to apply a number of newly developed techniques in harmonic analysis to the studyof Fourier multipliers. In a recent breakthrough, Guth--Wang--Zhang resolved the longstanding L^4 cone squarefunction conjecture and local smoothing conjecture for the wave equation in 2 spatial dimensions. One difficultyin proving estimates for the cone square function is that the setup is not directly amenable to the Lorentzrescaling, which features prominently in powerful induction-on-scale techniques which pervade modernharmonic analysis. To circumvent this issue, a new kind of estimate was introduced which subsumes thesquare function as a special case. This general framework is more robust and, in particular, is stable underLorentz transformation. There are many other examples of problems in which the scaling structure breaks downin a similar fashion, and it would be interesting to attempt to apply these methods, for instance, in the study ofBochner--Reisz-type multipliers associated to curves. In another direction, it is known that the local smoothingconjecture is closely related to the radial multiplier conjecture, which aims to characterise the L^p boundednessof radial Fourier multipliers in terms of the finiteness of the L^p norm of the associated kernel. It is natural toinvestigate whether advances in the understanding of local smoothing yield any new insights into radialmultipliers. These questions are also related to the incidence geometry of circular annuli, as expounded inworks of Heo--Nazarov--Seeger and Cladek. Recent advances in incidence geometry, such as the developmentof the polynomial partitioning method, may have some relevance to the investigation.