Inequalities in Probability and Harmonic Analysis

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Proposal: DMS-9870026 Principal Investigator: Stephen J. Montgomery-Smith Abstract: This proposal contains two projects. First, the principal investigator plans to investigate tail distributions of sums of random variables. Most results in the past have focused on asymptotic answers, that is, they obtain formulae that are valid, for example, when the number of variables summed is very large (the Law of Large Numbers, or the Central Limit Theorem). It is intended to find formulae that work in all situations. The second project concerns obtaining an inequality in which we wish to show that rank-one functions are quasi-convex. This would have profound consequences in harmonic analysis and the study quasi-conformal functions. It would settle the conjecture of Iwaniec concerning the best constant for the norm of the Beurling-Ahlfors Operator. It also has possible consequences in obtaining weak type estimates for singular integrals independent of dimension, which would be something quite unexpected in harmonic analysis. Finally, the research would find applications in applied mathematics, in problems dealing with homogenization of non-linear dielectrics. The first project deals with the following problem. If we are given a number of random quanitities, what can we say about their sum? For example, what can we say about the average height of ten people picked randomly from the population? This problem is fundamental in probability theory, tracing its historical roots back to the very beginnings of the discipline. There are still many unanswered problems in this area. The second project studies the nature of the concept of convexity. This is a cornerstone property of much mathematics, and it has been used in many areas of science, such as physics and engineering. This particular project studies quasi-convexity, a notion that emerged out of attempts to understand how elastic bodies are distorted under various forces and stresses. The principal investigator (along with D. Talbot) has already used t hese ideas to study problems of a different kind - namely, problems in electrostatics.