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：
2016
资助国家：
加拿大
起止时间：
2016-01-01 至 2017-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614301
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.

申请人建议继续研究围绕 1) 表征奇异积分算子的两个权重不等式，以及 2）研究更高维度的插值和电晕问题， 3）发展了次椭圆偏微分正则理论 Monge-Ampere 等方程和非线性波动方程， 4) 研究多项式增长度量的调和分析问题。这些研究中的主要问题通过多学科方法得到解决，该方法汇集了分析、几何、代数、组合学和算子理论的思想。两个例子包括高维 Drury-Arveson 空间的电晕问题的解决方案和希尔伯特变换的两个权重不等式的解决方案。电晕问题是通过偏微分方程中 d 杆问题的解、全纯函数的几何、形式代数、流氓因子的组合以及球上正算子的估计来解决的。双权重问题通过哈尔分解、希尔伯特变换的几何学以及相关二元多尺度分析的代数和组合学之间的相互作用来解决。我们建议继续研究解决高维有界解析函数著名的电晕问题以及高维 Riesz 变换和相关算子的两个权重不等式。