Collaborative research: Non-homogeneous harmonic analysis, two weight estimates and spectral problems.

PI建议集中精力解决分析和光谱理论中的几个经典问题，由于缺乏适当的技术工具，这些问题在过去20-50年里一直没有解决。这些问题包括：*高维分析能力模拟的bilipschitz等价性；*希尔伯特变换的两个权重估计；*离散薛定谔算子逆散射问题的适定性，即反非线性傅立叶变换的唯一性；*选定的非对易调和分析问题尽管这些问题跨越了几个不同的分析和数学物理领域，但我们最近的研究揭示了所提出的问题之间的显著联系。简而言之，它们都是统一的，因为它们都出现了相同类型的奇异核(通常是柯西核)。此外，这些问题都有相同的困难，内核被几乎任意函数(权重)的乘法“破坏”了。非齐次调和分析的最新发展使所提出的问题的成功解决成为可能。调和分析通过将复杂过程表示为具有良好行为的基本过程(正弦波、小波)的和来研究复杂过程。现代调和分析的核心部分是处理一种或另一种类型的“奇异积分算子”。这类算子在科学领域中无处不在：它们出现在数学物理、概率、工程、图像处理等领域。虽然奇异积分算子的理论现在已经发展得很好(始于Calderon和Zygmund的工作，并在他们之后的许多研究人员继续)，它处理定义在一个良好的“光滑”集上的算子，就像通常的欧几里得空间一样。然而，在许多问题中，人们需要在“坏”集上研究这类算子，非齐次调和分析是由PI引入的，用来准确地处理这样的情况：X.Tolsa最近对著名的解析能力次基问题的解是这种PI的非齐次分析理论的最重要的应用之一。Pi提出要解决几个经典问题，其中非齐次调和分析的框架自然而然地出现了。

ABSTRACT.PI's propose to concentrate their efforts on several classical problems inAnalysis and Spectral Theory that remained unsolved for the last 20--50years, due to the lack of appropriate technical tools. Among the problemsare:* bilipschitz equivalence for higher dimensional analogues of analyticcapacity; * two weight estimates for the Hilbert Transform;* well-posedness of the inverse scattering problem for the discreteSchrodinger operator,i.e., uniqueness of the inverse nonlinear Fourier transform;* selected problems of noncommutative harmonic analysisAlthough the problems span several different areas of analysis andmathematical physics, our recent research revealed striking connectionsbetween the proposed problems. To put it briefly, they all are unified bythe fact that in all of them the same type of singular kernels (usually theCauchy kernel) appears. Also, the problems share the same difficulty, thekernel got "spoiled'' by multiplication by virtually arbitrary functions(weights). Recent developments in the non-homogeneous harmonic analysis,which treats exactly this type of situations, made successful solution ofthe proposed problems plausible.Harmonic analysis investigates complex processes by representing them as asum of elementary ones (sinusoidal waves, wavelets) with well understoodbehavior. A central part of modern harmonic analysis deals with "singularintegral operators" of one type or another. Such operators are pervasive inthe scientific landscape: they turn up in mathematical physics, probability,engineering, image processing, etc. While the theory of singular integraloperators is now well developed (starting with works of Calderon and Zygmundand continued by numerous researchers after them), it deals with theoperators defined on a nice "smooth" set, like the usual Euclidean space.However, in many problems one needs to investigate such operators on a "bad"set, like surfaces with singularities and even on more pathological sets.The non-homogeneous harmonic analysis was introduced by the PI's to dealexactly with such situations: recent solution by X. Tolsa of the famoussubbaditivity problem for the analytic capacity is one of the mostimpressive applications of this PI's theory of nonhomogeneous analysis. PI'spropose to attack several classical problems, where the framework of thenon-homogeneous harmonic analysis appear naturally.