Date | Aug 08, 2024 |
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Speaker | Dominique Maldague |

Dept. | University of Cambridge |

Room | 27-116 |

Time | 17:00-19:00 |

Wave packet decomposition allows us to express functions with restricted frequency support as a superposition of wave packets (simpler functions which are localized in both space and frequency). A heuristic understanding of wave packets helps to explain the ideas behind proofs in Fourier restriction theory, and it is useful to think about examples in terms of the possible behavior of wave packets. I will explain the significance of a new type of inequality called a wave envelope (W.E.) estimate, which provides detailed information about the possible overlap patterns of wave packets that maximize the L^p norm. W.E. estimates were first introduced in the work of Guth-Wang-Zhang (GWZ) proving the sharp L^4 square function estimate for the cone in R^3. Guth-Maldague have since further refined the W.E. inequalities for the parabola and the cone in R^3, and used W.E. intuition in their square function estimate for the moment curve in R^n (although W.E.s are not explicitly mentioned there). Applications of W.E. estimates include sharp small cap decoupling estimates for the cone, new estimates for the size of exceptional sets in the 3D restricted projections problem, and a sharp multiplier-type problem for the moment curve.

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