Topics in harmonic analysis

近年来，调和分析及其相关领域(如偏微分方程组、组合学、数论和数学物理)的界面上出现了一阵活跃，产生了美丽和互惠互利的结果。调和分析中的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、偏微分方程式和数学物理中令人兴奋的结果。在这项提案中，申请人概述了调和分析的三个领域的未来目标：(A)圆锥的傅里叶估计及其应用，(B)线性多面体和单项多面体的分析，以及(C)具有退化相位的振荡积分算子。项目1和2解决了一类可以使用Wolff的光锥微局部估计来解决的问题(在他证明局部平滑猜想时出现)，并回答了关于平均和傅立叶积分算子和偏微分方程的各种问题。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3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，并将对几个复杂变量产生重大兴趣。项目4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(关于简并的傅立叶积分算子)有很深的联系，而且与奇点分解的极其丰富的理论也有很深的联系。

Recent years have seen a burst of activity in the interface of harmonic analysis and its related fields (such as partial differential equations, combinatorics, number theory, and mathematical physics), with beautiful andmutually 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, partial differential equations and mathematical physics. In this proposal, the applicant outlines future goals in three areas of harmonic analysis: (a) Fourier estimates for cones and their applications, (b) analysis of linear and monomial polyhedra, and (c) oscillatory integral operators with degenerate phases.Projects 1 and 2 address a class of problems that may be approached using Wolff's microlocal estimate forlight cones (arising in his proof of the local smoothing conjecture), and would answer a variety of questionsregarding 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 3 would mark nontrivial progress on a much-studied open question on Bergman kernel estimates, and would be of significant interest in several complex variables. Project 4 addresses a problem of which verylittle is known at the moment, but which has deep ties not only to numerous questions in harmonic analysis (regarding degenerate Fourier integral operators), but also to the extremely rich theory of resolution of singularities.