Mathematical Sciences: Fourier Analysis and Partial Differential Equations

数学科学：傅里叶分析和偏微分方程

• 批准号: 9007599
• 负责人: Robert Fefferman
• 金额: $ 19.05万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1993-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9007599&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Fourier; Analysis; Partial

Mathematical research on problems involving the application of harmonic analysis to partial differential equations will be carried out under this award. The work falls into three groups, the first of which concerns the Zygmund conjecture on multiple parameter families of rectangles. It was long believed that the k-parameter maximal function should have boundedness properties depending only on the parameter k and not on the dimension of the underlying space. That this was not necessarily the case was shown by F. Soria in 1986. Since this maximal function is central to any multi-dimensional differentiation theory, it is still of interest to find what the correct formulation of the bound on the maximal function should be. A second line of investigation concerns product singular integrals, a theory which developed from the one-variable theory only after the investment of considerable effort (the natural one-dimensional generalizations failed to carry over). Work is now progressing on the goal of showing that singular integrals can be expressed as pointwise limits of multiply truncated singular integrals. Finally, work in partial differential equations will concentrate on elliptic equations defined in a ball in Euclidean space. If the coefficients of the elliptic part of the operator are bounded and measurable, and the boundary values are required to be in one of the Lebesgue spaces, there are (somewhat surprisingly) cases where the equation has no solution. The question of existence has attracted the attention of many mathematicians over the past decade. What is now evolving are methods of determining existence in terms of the proximity of the operator to other operators for which the Dirichlet problem can be solved. This work will continue along the same line except that the domain under consideration will be unbounded. Significant differences arise in this setting, even for elliptic operators with smooth coefficients.

本奖项将对调和分析在偏微分方程式中的应用问题进行数学研究。工作分为三组，第一组涉及多参数矩形族上的Zygmund猜想。长期以来，人们一直认为k参数极大函数应该具有仅依赖于参数k而不依赖于基础空间的维度的有界性。F·索利亚在1986年表明，情况不一定是这样。由于这一极大值函数是任何多维微分理论的中心，寻找关于极大值函数的界的正确公式仍然令人感兴趣。第二条线是关于乘积奇异积分的，这是一种仅在投入了相当大的努力(自然的一维推广未能延续)后才从单变量理论发展起来的理论。现在的工作是证明奇异积分可以表示为多个截断的奇异积分的逐点极限。最后，偏微分方程的工作将集中在欧几里德空间中定义为球的椭圆型方程上。如果算子的椭圆部分的系数是有界的和可测的，并且边值被要求在一个勒贝格空间中，则存在(有点令人惊讶的)方程无解的情况。在过去的十年里，存在的问题引起了许多数学家的注意。现在正在发展的是根据算子与其他算子的接近程度来确定是否存在的方法，对于这些算子，狄里克莱特问题可以解决。这项工作将继续沿着同一条线进行，只是所考虑的领域将是无界的。在这种设置下，即使对于具有平滑系数的椭圆算子，也会出现显著的差异。