Harmonic Analysis, Function Theory and Partial Differential Equations

申请人建议继续研究围绕 1)刻画了奇异积分算子的两个权不等式， 2)研究更高维度的插值和电晕问题， 3)发展了次椭圆型偏微分的正则性理论 方程，如Monge-Ampere和非线性波动方程， 4)研究多项式增长测度的调和分析问题。 主要问题是克服这些调查通过多学科的方法汇集的想法，从分析，几何，代数，组合学和算子理论。两个例子包括解决的电晕问题的Drury-Arveson空间在高维和解决的两个重量不等式的希尔伯特变换。电晕问题的解决是使用偏微分方程中的d杆问题的解决方案，全纯函数的几何，形式代数，流氓因子的组合学和球上正算子的估计。两个权重的问题解决了哈尔分解之间的相互作用，希尔伯特变换的几何，以及相关的并矢多尺度分析的代数和组合学。我们建议继续研究解决著名的冠问题的有界解析函数在高维和两个权重不等式Riesz变换和相关的运营商在高维。

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.