Harmonic Analysis and Linear Elliptic Equations

本文研究奇异积分、权理论及其在二阶和高阶线性椭圆型偏微分方程解中的应用。目前关注的主要领域有三个。首先，我们研究与二阶椭圆算子相关的椭圆测度--既有散度形式，也有非散度形式。其次，我们研究了非光滑区域上一类高阶椭圆算子解的边值问题。这项建议的第三部分涉及与某些乘积域有关的极大函数和奇异积分-这些算子涉及各向同性扩张，结果需要研究由这种扩张结构产生的矩形的几何。在本世纪中叶，分析中的革命性思想在两个方面形成：调和分析中的奇异积分理论(Calderon-Zygmund分析学派)和椭圆方程理论(Di Giorgi，J.Nash)。每个子课题都提供了引人入胜的应用--第一个是应用调和分析/小波/傅立叶分析，第二个是非线性偏微分方程组理论，具有巨大的应用价值。但在这些研究的开始，在大约20年的时间里，每个受试者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S以来，这些领域之间的联系得到了越来越多的开发和了解，这对调和分析和偏微分方程领域都有很大的好处。本文作者的研究是在调和分析领域，而不是强调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.