Topics in Phase Plane Analysis

提案编号：DMS-0200241PI：Michael Lacey 摘要研究将在相平面分析的几个方面进行。具体问题来自三个不同的领域。一是研究E.M.斯坦因对卡尔森定理的某种扩展的猜想。这一猜想试图将卡尔森置于更大的背景中。 将此定理视为相位函数所有线性选择的上界。斯坦因的猜想涉及相位函数的所有多项式选择的上界，但多项式的次数是固定的。这个问题的进展已经表明，考虑卡尔森定理的某些离散类似物可能会富有成效。 在这里，数论的算术方面（以哈代-利特尔伍德圆法为例）应该占主导地位。 第三个领域是继续研究光滑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：同时研究时间变量和振荡变量中的函数。 过去在该领域取得的成功扩大了合成和分析技术的范围，可用于解决广泛的问题。对于双线性希尔伯特变换和卡尔森定理尤其如此。这些新问题比过去解决的问题要微妙得多，因此我们希望开发的新技术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.