Mathematical Sciences: Problems in Trigonometric Series and Applications

这个项目的工作将集中在一个和几个变量的谐波分析的经典问题上。它继续了三角级数的早期研究及其在高维振荡积分的调和分析和指数和中的应用。在后面的工作中，将特别强调周期偏微分方程和狄利克雷和的估计。振荡积分以多种方式进入数学分析。它们被认为是与傅里叶变换形式相同的变换，除了相位，它是线性的，被两个变量的函数所取代，空间和相位。我们还会研究所谓的集它将正交族的子集构造的函数的范数与系数的范数联系起来。在希尔伯特空间中，这些问题都是微不足道的，但是一旦转到其他范数，即使使用简单的三角函数，问题也会变得非常困难。一个新的推力涉及涉及非光滑边界条件的偏微分方程的柯西问题的研究。值得关注的是，当前人们感兴趣的某些基本偏微分方程（例如非线性薛定谔方程）在存在粗糙边界的情况下是否能得到良好的定姿。所研究的问题直接关系到调和分析和偏微分方程研究人员所面临的一些基本问题。在解析数论和几何测度论等不同领域的应用就不那么明显了。＊＊＊

9308345 Bourgain Work on this project will focus on classical problems of harmonic analysis in one and several variables. It continues earlier studies of trigonometric series and their applications to the harmonic analysis of higher dimensional oscillatory integrals, and to exponential sums. In the latter work, special emphasis on periodic partial differential equations and estimates of Dirichlet sums will be carried out. Oscillatory integrals enter into mathematical analysis in many ways. They are recognized as transforms identical in form to the Fourier transform except that the phase, which is linear, is replaced by a function of two variables, space and phase. Work will also be done on the so-called lambda-p sets which relate the norms of functions constructed by subsets of orthogonal families with the norms of the coefficients. In the Hilbert space setting the questions are trivial, but once one goes over to other norms, even using simple trigonometric functions, the problems become very difficult. A new thrust involves studies of the Cauchy problem in partial differential equations involving nonsmooth boundary conditions. Of concern is whether or not certain of the fundamental partial differential equations of current interest (nonlinear Schrodinger, for example) are well-posed in the presence of rough boundaries. The questions under investigation have direct bearing on some of the fundamental issues facing researchers in harmonic analysis and partial differential equations. Not so obvious are applications to such diverse areas as analytic number theory and geometry measure theory. ***