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. ***
