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工作者感兴趣的四个问题。第一个问题是求解Beurling-Ahlfors算子B的Lebesgue p-范数的计算。B在拟共形映射理论中占有重要的地位，这个特殊的问题有一个著名的Iwaniec猜想，大约25年前，当p大于等于2时，算子范数是p-1。目前的技术，涉及到由于伯克霍尔德的鞅方法，达到了2（p-1）的上限，而R. Banuelos和PI的最近工作（在本提案发出后）进一步将其降低到1.58（p-1）以下。下一个目标是求出外常数的平方根或者证明这个猜想。第二个问题是找出正交调和函数的弱型（p, p）常数（p2）。作为一个重要的特例，求出最佳常数C_p使得给定单位圆上任意函数f p范数等于1，圆上f的共轭函数的绝对值超过任意正t0的点的集合的度量小于等于(C_p)/（t^p）PI最近在一般情况下解决了同样的问题，当p=1时，扩展了已知的结果。剩下的两个问题是将已知算子的弱型不等式推广到Radon奇异测度的一般类，以及寻找与几何测度理论问题的潜在联系。算子，比如T，作用于一个奇异测度v， v支持在R^n的子集上，例如一个k维Lipschitz图L。当L接近时，得到的函数Tv具有渐近行为，可以计算出来，并由此可以推测出相应的弱型（p, p）不等式。事实上，除了常数依赖于l的Lipschitz常数之外，它是可以被证明的。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，在项目的最后一部分，PI提出探索弱型不等式的调和分析与几何测度理论之间的某些有趣联系。这项工作具有重要的智力价值，因为它涉及到相互联系不同数学领域的问题。前两个问题有着吸引分析师和概率学家的奇妙历史，导致了这两个领域新技术的发展和应用，并强烈表明更深层次的相互联系尚未被发现。第二部分讨论了算子在奇异测度上的作用，在分形的几何测度理论中给出了令人着迷的猜想。因此，这一建议的核心是在明显的个性背后寻找统一，这是科学的普遍动机。至于具有更广泛影响的特定实例，可以考虑分形，这是自然出现在物理科学和数学中的东西。在上一个问题中，PI表明分形的距离管的体积可以通过谐波分析中的算符来理解。因此，如果这些运算符的数学得到发展，那么对分形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.