Harmonic Analysis, Function Theory and Partial Differential Equations

申请人建议继续围绕*1)刻画奇异积分算子的两个权重不等式，以及*2)研究高维的插补和日冕问题，*3)发展次椭圆偏微分方程如Monge-Ampere方程和非线性波动方程的正则性理论，*4)研究多项式增长测度的调和分析问题。*这些研究中的主要问题是通过结合分析、几何、代数、组合学和算子理论的多学科方法来克服的。两个例子包括高维Drury-Arveson空间的日冕问题的解和希尔伯特变换的两个权不等的解。利用偏微分方程组中d-bar问题的解、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来解决日冕问题。通过Haar分解、Hilbert变换的几何以及相关的并元多尺度分析的代数和组合学之间的相互作用，解决了两个权重问题。我们建议继续研究解决著名的高维有界解析函数的冠状问题和高维Riesz变换及其相关算子的两个权不等。

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.**