Topics in Phase Plane Analysis

项目编号：DMS-0200241PI: Michael lacey摘要将对相平面分析的几个方面进行研究。具体的问题来自三个不同的领域。一是研究E.M. Stein关于Carleson定理的某种推广的一个猜想。这一猜想试图将卡尔森置于一个更大的背景中。把这个定理看作是所有相函数的线性选择的最高定理。斯坦的猜想涉及到所有相函数的多项式选择的上极值，然而多项式的度数是固定的。在这个问题上的进展已经表明，考虑Carleson定理的某些离散类似物可能是富有成效的。在这里，数论的算术方面，如哈代-利特尔伍德圆法的例子应该占主导地位。第三个领域是继续研究光滑直线族上希尔伯特变换的有界性问题。这些研究旨在加深我们对相平面分析的理解：同时研究时间和振荡变量中的函数。过去在这一领域的成功导致了综合和分析技术的扩大，这些技术可以用于解决范围广泛的问题。对于双线性希尔伯特变换和Carleson定理尤其如此。这些新问题比过去解决的问题要微妙得多，因此我们寻求开发的新技术希望比以前的技术更强大。

Proposal Number: DMS-0200241PI: Michael LaceyABSTRACTResearch will be conducted on several aspects of phase planeanalysis. The specific questions come from three different areas. One is to study a conjecture of E.M. Stein on a certain extension of Carleson's theorem. This conjecture seeks to placeCarleson in a larger context. Consider this theorem as a supremum over all linear choices of phase functions. Stein's conjecture concerns a supremum over all polynomial choices of phase functions, with however the degree of the polynomial fixed. Advances on this question already suggestthat it might be fruitful to consider certain discrete analogues of Carleson's theorem. Here, arithmetic aspects of Number Theory, as exemplified by the Hardy-Littlewood Circle Method should predominate. A third area iscontinued investigation into the question of the boundedness of the Hilbert transform on smooth families of lines. These investigations seek to deepen our understanding of phase plane analysis: The study of functions in the temporal and oscillatory variables simultaneously. Past successes in this area have lead to a broadening of the synthetic and analytic techniques that can be brought to bear on a wide range of questions. This is especially true for thebilinear Hilbert transform and Carleson's theorem. These new questions are far more subtle than those addressed in the past, and so the new techniques that we seek to developare hoped to be more powerful than previous ones.