Topics in Harmonic Analysis

该提案涉及谐波分析的三个子领域：(1)锥乘子和局部平滑，(2)振荡积分和积分算子，以及(3)薛定谔算子的频谱分析。(1)中的工作使用了Wolff在与光锥相关的傅里叶分析估计理论中的一个重要结果。可能的应用领域包括(a)与空间曲线相关的乘子，(b)将局部平滑推广到傅里叶积分算子的特殊类别，(c)与空间曲线相关的最大平均值的局部平滑，以及(d) kakeya型集的Hausdorff维数。(2)中的项目处理高维（大于2）中的退化振荡积分和积分算子的特殊情况，已知这些情况表现出二维对应项中所没有的特征。分析机制是由Phong和Stein为二维情况开发的，但得到的结果具有非常不同的性质。这里的长期目标是设计一种可解析的方法来解决奇点。在Carbery， Wainger和Wright之后，本工作的另一部分集中在沿多项式曲面的二重希尔伯特变换上。(3)中的项目是理解具有矩阵值势的薛定谔算符的谱理论的一部分。这借鉴了Guillope和Zworski， Laptev和Weidl以及Korotyaev的早期工作。以下是对上述项目的非技术描述，并简要说明其在科学学科中的适用性。(1)中的项目可以看作是对非均匀介质（用于地震成像）中的波传播的研究。“局部平滑”量化了作为时空函数的传播波与在固定时间内单独作为空间函数的传播波的规律性增益。这个问题的分支在平面上的圆的排列上有组合的味道，正如Wolff指出的那样。(2)中的项目涉及在偏微分方程的解中出现的积分和积分算符，包括热方程、波动方程和Korteweg-deVries方程等基本方程。这些积分的一个重要特征是在被积函数中存在复值指数因子。在我们研究的例子中，指数通常是一个多项式，在某一点高度消失。多项式的“消失顺序”决定了积分的衰减速率。分析这些物体的一个主要成分是代数几何中的一种技术，称为奇点分解——一种分解多项式并研究其根的系统方法。薛定谔算符的谱理论是(3)中项目的基础，它与量子力学密切相关，量子力学在物理学和许多工程科学学科中有着广泛的应用。目前的项目是了解薛定谔算符在大于1的维度上的光谱特性，更具体地说，是计算这些算符的共振数。共振是束缚态或特征值概念的概括，在电子运动的退出时间方面具有物理意义。这些项目的跨学科性质已经证明是申请人非常有益的研究经历。

The proposal addresses three subfields of harmonic analysis : (1) cone multipliers and local smoothing, (2) oscillatory integrals and integral operators, and (3) spectral analysis of Schroedinger operators. The work in (1) uses an important result of Wolff in the theory of Fourier-analytic estimates associated to the light cone. Possible areas of application include (a) multipliers related to space curves, (b) generalizations of local smoothing to special classes of Fourier integral operators, (c) local smoothing of maximal averages associated with space curves, and (d) Hausdorff dimension of Kakeya-type sets. The projects in (2) deal with special cases of degenerate oscillatory integrals and integral operators in high dimensions (larger than two) that are known to exhibit features absent in their two-dimensional counterparts. The analytic machinery is that developed by Phong and Stein for the two-dimensional case, but the results obtained are of a very different nature. A long-term goal here is to devise an analytically accessible method of resolution of singularities. Another part of this work concentrates on the double Hilbert transform along polynomial surfaces, following Carbery, Wainger and Wright. The projects in (3) are part of an effort to understand the spectral theory of Schroedinger operators with matrix-valued potentials. This draws on earlier work of Guillope and Zworski, Laptev and Weidl, and Korotyaev. The following is a more nontechnical description of the projects outlined above, with a brief note about applicability in scientific disciplines. The projects in (1) may be viewed as a study of wave propagation in non-uniform media (as used in seismic imaging). ``Local smoothing'' quantifies the gain in regularity of the propagating wave viewed as a function of space-time compared with the same wave considered as a function of space alone for a fixed time. Offshoots of this problem have combinatorial flavors in terms of arrangements of circles in the plane, as pointed out by Wolff. The projects in (2) deal with integrals and integral operators which come up in solutions to partial differential equations, including fundamental ones like the heat equation, wave equation and Korteweg-deVries equation. An important feature of these integrals is the presence of a complex-valued exponential factor in the integrand. In the case under study, the exponent is typically a polynomial that vanishes to high degree at a point. The ``order of vanishing'' of the polynomial contributes to the rate of decay of the integrals. A main ingredient in the analysis of these objects is a technique from algebraic geometry known as resolution of singularities -- a systematic method for factorizing polynomials and studying their roots. The spectral theory of Schroedinger operators, which is the basis for the projects in (3), is intimately related to quantum mechanics, which finds vast applications in physics and many disciplines of the engineering sciences. The current project is to understand the spectral properties of the Schroedinger operator in dimensions larger than one, and more specifically to count the number of resonances of such operators. Resonances are generalizations of the concept of bound states or eigenvalues, and have a physical significance in terms of exit times in electron motion. The interdisciplinary nature of these projects has proved an extremely rewarding research experience for the applicant.