Sparse Fourier restriction for the cone

Abstract

In Fourier restriction theory, weighted inequalities allow us to probe the shape of level sets. In this thesis, we describe a new weighted Fourier extension estimate for the cone and its connection with the Mizohata–Takeuchi conjecture. The main result Theorem 3.1 builds on techniques from geometry originally explored by Tom Wolff in this context. The proof uses circular maximal function estimates first proved by Wolff and later generalized by Pramanik–Yang–Zahl in their work on restricted projections as a black box.