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

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Project Abstract This 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< p<2, 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.