Topics in Harmonic Analysis

近年来，在谐波分析及其相关领域(如偏微分方程组、组合学、数论和数学物理)的界面上出现了大量的活动，产生了美丽和互惠互利的结果。调和分析中长期悬而未决的问题已经用组合技术解决了，而调和分析的工具帮助证明了数论和数学物理中令人兴奋的结果。在这个项目中，PI概述了调和分析的三个领域的未来目标：(A)锥的傅立叶估计及其应用(第二节)，涉及Wolff的局部平滑估计(B)线性和单项多面体的分析，及其在(I)相关的极大算子和(Ii)弱伪凸域的Bergman核估计中的应用(第三节)，以及(C)具有退化相位的振荡积分算子的衰减性质)。项目1、2和3解决了一类可以用Wolff的光锥的微局部估计来接近的问题(在他的局部平滑猜想的证明中出现)，并将回答有关平均、傅里叶积分算子和偏微分方程式的各种问题。他们需要更深入地理解和扩展在他的证明中使用的微妙的组合论点。项目4的结果将标志着关于Bergman核估计的一个被广泛研究的公开问题取得了不平凡的进展，并将对几个变量的复杂分析产生重大兴趣。项目5解决了一个目前知之甚少的问题，但这个问题不仅与调和分析中的许多问题(关于退化的傅立叶积分算子)有很深的联系，而且与代数几何的一个极其丰富和繁荣的领域，即奇点的分解有很深的联系。

ABSTRACTRecent years have seen a burst of activity in the interface of harmonicanalysis and its related fields (such as partial differential equations, combinatorics, number theory, and mathematical physics) with beautiful and mutually beneficial consequences. Long-standing open problems in harmonic analysis have been solved using combinatorial techniques, while tools from harmonic analysis have helped in proving exciting results in number-theory and mathematical physics. In this project, the PI outlines future goals in three areas of harmonic analysis: (a) Fourier estimates for cones and their applications (section 2), involving a local smoothing estimate of Wolff (b) analysis of linear and monomial polyhedra, and their applications to (i) associated maximal operators and (ii) Bergman kernel estimates for weakly pseudoconvex domains (section 3), and (c) decay properties of oscillatory integral operators with degenerate phases section 4).Projects 1, 2 and 3 address a class of problems that may be approachedusing Wolff's microlocal estimate for light cones (arising in his proof of the local smoothing conjecture), and would answer a variety of questions regarding averaging and Fourier integral operators and partial differential equations. They necessitate a deeper understanding and extension of the delicate combinatorial arguments employed in his proof. Results from project 4 would mark nontrivial progress on a much-studied open question on Bergman kernel estimates, and would be of significant interest in complex analysis of several variables. Project 5 addresses a problem of which very little is known at the moment, but which has deep ties not only to numerous problems in harmonic analysis (regarding degenerate Fourier integral operators), but also to an extremely rich and thriving area of algebraic geometry, namely resolution of singularities.