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

摘要.PI 建议将精力集中在分析和谱理论中的几个经典问题上，这些问题由于缺乏适当的技术工具而在过去 20--50 年间仍未得到解决。 其中的问题包括： * 分析能力的高维类似物的 bilipschitz 等价； * 希尔伯特变换的两个权重估计；* 离散薛定谔算子的逆散射问题的适定性，即逆非线性傅里叶变换的唯一性；* 非交换调和分析的选定问题虽然这些问题跨越了几个不同的分析领域和数学物理，但我们最近的研究揭示了所提出的问题之间的惊人联系。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，这些问题也有同样的困难，内核被几乎任意函数（权重）相乘所“破坏”。非齐次调和分析的最新发展恰好处理了此类情况，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理解行为的基本过程（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涉及“奇异积分算子”这种或那种类型的算子在科学领域中很普遍：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很发达（从卡尔德隆和齐格蒙德的工作开始，后来许多研究人员继续进行），但它处理的是在一个很好的“光滑”集合上定义的算子，就像通常的欧几里得空间一样。然而，在许多问题中，我们需要在一个“坏”集合上研究这样的算子，比如曲面。 PI 引入了非齐次调和分析来准确地处理此类情况：X. Tolsa 最近对著名的分析能力次弱性问题的解决方案是该 PI 的非齐次分析理论最令人印象深刻的应用之一，PI 提出解决几个经典问题，其中非齐次调和分析的框架自然出现。

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.