Bilinear Singular Integrals

摘要 莱西DMS-9706884 Lacey与C.蒂勒，已经取得了实质性的 阿尔贝托·卡尔德隆猜想的解决进展 关于双线性希尔伯特变换的有界性 从L^2乘以L^2到L^1。 莱西和蒂勒将致力于发展一个更完整的理解理论的双线性奇异积分。 他们的证明方法可以细化到三个重要方面， 谐波分析可以从同一组 基本原则。 它们是Lipschitz上的柯西积分 曲线，Carleson点态收敛定理 傅里叶级数和双线性奇异积分和极大 功能协调发展的证明的方法是基于一种新的使用， 相平面技术 将探讨这些技术在其他问题上的应用。 一个乐谱是一个音乐的代表性， 时间和频率方面。 相平面分析是一种 适应性的乐谱。这类人的特点是 一个分析正是一个乐谱，一个高度 时间和频率变量的局部化。 这项建议 从事相平面分析及其应用的研究 到谐波分析中的运算符， 难以计算 因为它们的非本地性谐波分析为信号处理提供了数学基础，最近又因小波理论而普及。

ABSTRACT Lacey DMS-9706884 Lacey, in collaboration with C. Thiele, has made substantial progress towards settling the conjecture of Alberto Calderon concerning the boundedness of the bilinear Hilbert transform from L^2 times L^2 into L^1. Lacey and Thiele will work on developing a more complete understanding of the theory of bilinear singular integrals. Their method of proof can be refined to the point at which three significant aspects of harmonic analysis can be understood from the same group of basic principles. They are the Cauchy integral on Lipschitz curves, Carleson's Theorem on the pointwise convergence of Fourier series and bilinear singular integrals and maximal functions. The method of proof is based on a novel use of phase plane tecniques. The applications of these techniques to other questions will be explored. A musical score is a representation of the music built around temporal and frequency aspects. A phase plane analysis is an adaptive counterpart of the musical score. The hallmark of such an analysis is precisely that of a musical score, a high degree of localization in time and frequency variables. This proposal pursues the study of phase plane analysis and its applications to operators in Harmonic Analysis which are difficult to compute due to their non--local nature. Harmonic Analyisis provides the mathematical foundation for Signal Processing, most recently popularized by the theory of wavelets.