Iterated Fourier Series and Integrals

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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.