New developments in geometric Fourier analysis

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This project aims to harness the power of a variety of newly available tools in harmonic analysis to study classical objects such as geometric maximal functions. One possible direction is to study variants of Bougain's circular maximal function. This operator acts on functions on the Euclidean plane by taking maximal averages over concentric circles. It is intimatelyrelated to the behaviour of space/time averages of solutions to the linear wave equation. Recently, the local smoothing conjecture for the wave equation was established by Guth--Wang--Zhang. This conjecture implies (and is substantially stronger than) Bourgain's circular maximal function theorem, as well as many other classical results in harmonic analysissuch as the Bochner--Riesz and restriction conjectures in 2 dimensions. The proof of the local smoothing conjecture involves a powerful Littlewood--Paley square function inequality for functions frequency supported near the lightcone. This inequality, and the methods used to prove it, are likely to have a broad range of further applications and it is of great interest to explore other situations where they may apply.