Mathematical Sciences: Problems in Trigonometric Series and Applications

小行星9308345 这个项目的工作将集中在一个和几个变量的调和分析的经典问题。它继续早先的研究三角级数及其应用的谐波分析的高维振荡积分，指数和。 在后者的工作中，特别强调周期偏微分方程和估计的狄利克雷和将进行。 振荡积分以多种方式进入数学分析。 它们被认为是在形式上与傅立叶变换相同的变换，除了线性的相位被两个变量的函数，空间和相位代替。 工作也将做所谓的dupda-p集有关规范的职能建设的子集正交家庭的规范的系数。 在希尔伯特空间的设置问题是微不足道的，但一旦一个去到其他规范，即使使用简单的三角函数，问题变得非常困难。 一个新的推力涉及研究的柯西问题的偏微分方程涉及非光滑边界条件。 值得关注的是，当前感兴趣的某些基本偏微分方程（例如非线性薛定谔方程）在粗糙边界的存在下是否是适定的。 调查中的问题有直接关系的一些研究人员在调和分析和偏微分方程面临的基本问题。 不那么明显的是应用等不同领域的解析数论和几何测度理论。 ***

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. ***