Iterated Fourier Series and Integrals

数学物理中的一个基本问题是所谓的平面n体问题。广义地说，它要求人们分析几个粒子/物体的轨道，这些粒子/物体最初在平面上的一个固定点上独立地圆周运动，然后开始相互作用，可能以非线性的方式相互作用。感兴趣的基本问题如下：随着时间的推移，在什么情况下这些粒子/物体的轨道保持有界？例子包括太阳系，行星围绕太阳运动，以及（平面）星系，恒星围绕黑洞运动。最近，人们发现这个问题与谐波分析领域有着深刻的联系，本项目旨在更定量地探索和理解这些联系。有几个非常有趣的非线性算符需要从谐波分析的角度来理解，以便至少在某些特殊情况下对上面的问题给出有意义的答案。其中一些包括与由组合树生成的迭代傅立叶积分相关的carleson型极大算子。这些树的复杂性自然地与粒子/物体之间非线性相互作用的复杂性联系在一起。例如，二次交互生成二叉树，三次交互对应三叉树，等等。特别是，最简单的一叉树是线性相互作用的树，几年前，首席研究员与Tao和Thiele合作研究了这种树。这是产生标准迭代傅里叶级数和积分的情况。这个项目的目的是开发必要的分析工具来理解这些非线性算子的有界性。

A fundamental question in mathematical physics is the so-called planar N-body problem. In broad terms, it asks one to analyze the orbits of several particles/bodies that initially move independently and circularly around a fixed point in a plane, and then start to interact with each other, possibly in a nonlinear fashion. The basic question of interest is the following: With the passage of time, under what circumstances do the orbits of these particles/bodies remain bounded? Examples include the Solar System, where the planets are moving around the Sun, and the (planar) galaxies, where the stars move around a black hole. Recently, it has been discovered that there are deep connections between this problem and the field of harmonic analysis, and the present project aims to explore and understand these connections more quantitatively.There are several extremely interesting nonlinear operators that one needs to understand from a harmonic analysis point of view in order to give meaningful answers to the question above, at least in some particular situations. Some of them include Carleson-type maximal operators associated to iterated Fourier integrals that are generated by combinatorial trees. The complexity of these trees is naturally linked to the complexity of the nonlinear interactions between the particles/bodies. For instance, quadratic interactions generate binary trees, cubic interactions corresponds to ternary trees, and so on. In particular, the simplest case of 1-ary trees is the one that comes from linear interactions, and it was studied by the principal investigator in collaboration with Tao and Thiele some years ago. This is the case that generated the standard iterated Fourier series and integrals. The aim of this project is to develop the necessary analytical tools to understand the boundedness properties of such nonlinear operators.