Best Norm Constants and Weak-type Inequalities for Operators in Harmonic Analysis

项目摘要本提案包含分析和概率论专业人员感兴趣的四个问题。第一个问题是关于Beurling-Ahlfors算子B的Lebesguep-范数的计算。B在拟共形映照理论中占有重要的地位，并且这个问题有一个著名的Iwaniec猜想，大约25岁，对p大于或等于2，算子范数是p-1。由于Burkhold的方法，目前的技巧得到了2(p-1)的上界，R.Banuelos最近的工作(在这个建议提出之后)和PI进一步将这个上界降低到1.58(p-1)以下。下一个目标是得到外部常量的平方根2，或者证明猜想。第二个问题是求正交调和函数的弱型(p，p)常数(P2)。作为一种特殊的重要情况，求最佳常数Cp，使得给定单位圆上任意p-范数等于1的函数f，其共轭函数绝对值超过任意正t0的点集的测度小于或等于(Cp)/(t^p).PI最近在一般情况下解决了同样的问题，当1 p2时，推广了p=1的已知结果。剩下的两个问题涉及将著名算子的弱型不等式推广到一般的Radon奇异测度类，以及寻找几何测度论中问题的潜在联系。算符T作用于支撑在R^n的子集上的奇异测度v，例如k维Lipschitz图L。当逼近L时，得到的函数TV具有渐近行为，可以计算并由此猜想相应的弱型(p，p)不等式。事实上，这是可以证明的，除非这个常数将取决于L的Lipschitz常数。问题是要证明这个弱型常数独立于Lipschitz常数，这将允许推广。最后，在项目的最后部分，PI建议探索弱型不等式的调和分析与几何测量理论之间的某些有趣的联系。这项工作具有重要的智力优势，它涉及到相互关联的不同数学领域的问题。前两个问题有着吸引分析家和概率论者的美好历史，导致了这两个领域新技术的开发和应用，并强烈表明尚未发现更深层次的相互联系。该提案的第二部分涉及算子对奇异测量的作用，最终以分形几何测量理论中有趣的猜想告终。因此，这一提议的核心是在明显的个性背后寻求统一，这是科学的普遍动机。至于更广泛影响的特定实例，考虑一下分形学，这是自然科学和数学中自然产生的东西。在最后一个问题中，PI建议可以通过对调和分析中的算符的分析来理解距离管的体积关于一个分形体。因此，如果这些运算符的数学得到发展，那么对分形学的物理理解就会得到更好的理解，我们可能会将这种理解应用到处理分形学的物理科学中。

Project AbstractThis proposal contains four problems of interest to those who work in analysis and probability theory. The first problem addresses the computation of the Lebesgue p-norm of the Beurling-Ahlfors operator B. B has an important place in quasi-conformal mapping theory, and this particular problem has a well-known conjecture of Iwaniec, about 25 years old, that the operator norm is p-1, for p greater than or equal to 2. Present techniques, which involve martingale methods due to Burkholder, attain the upper bound of 2(p-1), and recent work (after this proposal was sent) of R. Banuelos and the PI has further reduced this to below 1.58(p-1). The next objective is either to get square root 2 for the outer constant or to prove the conjecture. The second problem is to find the weak-type (p, p) constant (p 2) for orthogonal harmonic functions. As a special important case, find the best constant C_p so that given any function f on the unit circle with p-norm equal to 1, the measure of the set of points on the circle where the absolute value of the conjugate function of f exceeds any positive t 0 is less than or equal to (C_p)/(t^p). The PI has recently solved the same problem in the general setting when 1 p2, extending known results for p=1. The remaining two problems deal with extending weak-type inequalities for well-known operators to general classes of Radon singular measures and finding potential connections to questions in geometric measure theory. The operator, say T, acts on a singular measure v supported on a subset of R^n, for instance a k-dimensional Lipschitz graph L. The resulting function Tv has asymptotic behavior as L is approached that can be calculated and from which a corresponding weak-type (p, p) inequality can be conjectured. In fact, it can be proved except the constant will depend on the Lipschitz constant of L. The problem is to show that this weak-type constant is independent of the Lipschitz constant, which will allow generalizations. Finally, in the last part of the project, the PI proposes to explore certain interesting connections between the harmonic analysis of weak-type inequalities and geometric measure theory.This work has the important intellectual merit that it involves problems that interconnect distinct areas of mathematics. The first two problems have a wonderful history of attracting both analysts and probabilists, resulting in the development and application of new techniques in both areas and strongly suggesting deeper interconnections yet to be found. The second part of the proposal, which deals with the action of operators on singular measures, culminates with fascinating conjectures in the geometric measure theory of fractals. Thus, at the heart of this proposal is the search for Unity behind apparent individualities, which is the universal motivation of science. As for a particular instance of broader impact, consider a fractal, something that naturally arises in physical sciences as well as in mathematics. The PI suggests in the last problem that the volume of a distance tube about a fractal can be understood through analysis of operators in harmonic analysis. Thus if the mathematics of these operators is developed, then the physical understanding of fractals is better understood, and we could potentially apply this understanding to the physical sciences which deal with fractals.