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

摘要：由于缺乏适当的技术工具，PI建议集中精力解决分析和谱理论中的几个经典问题，这些问题在过去的20- 50年里一直没有得到解决。 其中的问题是：* bilipschitz等价的高维类似物的分析能力; * 两个重量估计的希尔伯特变换;* 适定性的逆散射问题的离散薛定谔算子，即，非线性傅立叶逆变换的唯一性;* 非对易调和分析的选定问题虽然这些问题跨越了分析和数学物理的几个不同领域，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，这些问题都有同样的困难，内核被几乎任意的函数（权重）相乘而“损坏”了。最近的发展，在非齐次谐波分析，它处理正是这种类型的情况下，成功的解决方案提出的问题似是而非。谐波分析研究复杂的过程，将它们表示为一个基本的（正弦波，小波），以及understood的行为。现代调和分析的一个中心部分涉及一种或另一种类型的“奇异积分算子”。这样的算子在科学领域中无处不在：它们出现在数学物理、概率论、工程、图像处理等领域，而奇异积分算子的理论已经发展得相当成熟（从Calderon和Zygmund的工作开始，并在他们之后继续许多研究人员），它处理定义在一个很好的“光滑”集上的算子，就像通常的欧几里德空间。然而，在许多问题中，人们需要在一个“坏“的集合上研究这样的算子，比如具有奇异性的曲面，甚至在更病态的集合上。著名的解析容量次幂问题的求解是PI的非齐次分析理论的一个重要应用。PI提出了解决几个经典问题，其中非齐次谐波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.