Harmonic Analysis and Linear Elliptic Equations

摘要本文主要研究奇异积分、权重理论及其在二阶和高阶线性椭圆型偏微分方程中的应用。目前主要关注三个领域。首先，研究二阶椭圆算子的发散和非发散形式的椭圆测度。其次，研究了一类非光滑域上高阶椭圆算子解的边值。本建议的第三部分涉及与某些积域相关的极大函数和奇异积分-这些算子涉及各向同性膨胀，结果需要研究由这种膨胀结构产生的矩形族的几何形状。在本世纪中叶，分析中的革命性思想在两条战线上形成：调和分析中的奇异积分理论（卡尔德隆-齐格蒙德分析学派）和椭圆方程理论（迪·乔治，j·纳什）。每一门学科都提供了令人着迷的应用——第一门学科在应用谐波分析/小波/傅立叶分析等领域，第二门学科在非线性偏微分方程理论中，具有巨大的应用价值。但是在这些研究开始的时候，在20年左右的时间里，每个研究对象的发展都是独立进行的，几乎没有相互作用。然而，自20世纪80年代初以来，这些领域之间的联系得到了越来越多的开发和理解，这对谐波分析和偏微分方程领域有很大的好处。这个提议者的研究是在谐波分析的这个领域，重点是它与偏微分方程的联系和启发。

ABSTRACTThis proposal is concerned with research on singular integrals, the theory ofweights, with applications to linear elliptic partial differential equations of second and of higher order. There are three main areas ofcurrent focus. First, we study the elliptic measure associated toa second order elliptic operator - both divergence and nondivergence form. Second, we are studying the boundary values of solutions tocertain higher order elliptic operators in non-smooth domains. The thirdpart of this proposal concerns the maximal functions and singularintegrals associated to certain product domains - such operators involvenonisotropic dilations and results require a study of the geometry offamilies of rectangles arising from this dilation structure.In the middle of this century, revolutionary ideas in analysis werebeing forged on two fronts: the theory of singular integrals in harmonic analysis (the Calderon-Zygmund school of analysis) and thetheory of elliptic equations (Di Giorgi, J. Nash). Each of thesesubjects has provided fascinating applications - the first in suchareas as applied harmonic analysis/wavelets/Fourier analysis, and thesecond in the theory of nonlinear partial differential equations, oftremendous value in applications. But at the beginning of these studies, andfor a period of twenty or so years, the development of each of thesesubjects proceeded independently, and with little interaction. However,since the early 1980's, the connections between these areas havebeen increasingly exploited and understood, to the great benefit ofthe fields of harmonic analysis and partial differential equations.This proposer's research is in this field of harmonic analysis, withan emphasis on its connection to and inspiration from partial differential equations.