Topics in Phase Plane Analysis

提案编号：DMS-0200241 PI：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。一个是研究E.M. Stein关于Carleson定理的一个推广。这个猜想试图把Carleson放在一个更大的背景下。 考虑这个定理作为一个上确界在所有线性选择的相位函数。Stein的猜想关注的是相位函数的所有多项式选择的上确界，然而多项式的次数是固定的。在这一问题上的进展已经表明，考虑某些可能是富有成效的， Carleson定理的离散类似物。 在这里，数论的算术方面，例如哈代-利特尔伍德圆法应该占主导地位。 第三个领域是继续调查的问题的有界性的希尔伯特变换光滑家庭的线。这些调查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，可以对广泛的问题产生影响。这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.