Harmonic Analysis, Function Theory and Partial Differential Equations

调和分析、函数论和偏微分方程

基本信息

批准号：
RGPIN-2015-06688
负责人：
Sawyer, Eric
金额：
$ 1.24万
依托单位：
McMaster University
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2019
资助国家：
加拿大
起止时间：
2019-01-01 至 2020-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675832
关键词：
Harmonic Analysis Function Theory Partial

项目摘要

The applicant proposes to continue research into the circle of ideas surrounding***1) characterizing two weight inequalities for singular integral operators, and***2) investigating interpolation and corona problems in higher dimensions,***3) developing the regularity theory of subelliptic partial differential*equations such as the Monge-Ampere and nonlinear wave equations,***4) investigating harmonic analysis questions for polynomial growth measures.***Major problems are overcome in these investigations through a multidisciplinary approach bringing together ideas from analysis, geometry, algebra, combinatorics and operator theory. Two examples include the solution to the corona problem for the Drury-Arveson space in higher dimensions and the solution to the two weight inequality for the Hilbert transform. The corona problem was solved using solutions to the d-bar problem in partial differential equations, the geometry of holomorphic functions, the algebra of forms, the combinatorics of rogue factors and estimates for positive operators on the ball. The two weight problem was solved with an interplay between Haar decompositions, the geometry of the Hilbert transform, and the algebra and combinatorics of the associated dyadic multiscale analysis. We propose to continue investigations toward settling the famous corona problem for bounded analytic functions in higher dimensions and the two weight inequalities for Riesz transforms and related operators in higher dimensions.**

The applicant proposes to continue research into the circle of ideas surrounding***1) characterizing two weight inequalities for singular integral operators, and***2) investigating interpolation and corona problems in higher dimensions,***3) developing the regularity theory of subelliptic partial differential*equations such as the Monge-Ampere and nonlinear wave equations,***4) investigating harmonic analysis questions for polynomial growth measures.***Major problems are在这些调查中通过多学科的方法克服了分析，几何，代数，组合和操作者理论的思想。两个例子包括在更高维度的Drury-Arveson空间的电晕问题的解决方案，以及对希尔伯特变换的两个重量不等式的解决方案。使用针对D-BAR问题的解决方案，部分微分方程，圆形功能的几何形状，形式的代数，流氓因素的组合以及球上正算子的估计值来解决电晕问题。通过HAAR分解，Hilbert Transform的几何形状以及相关的二元多尺度分析的代数和组合，解决了两个重量问题。我们建议继续进行调查，以解决较高维度的有限分析功能的著名电晕问题，以及Riesz Transforms和相关操作员在较高维度上的两个重量不平等。**