The project is devoted to settling - in part or in full - of three long-standing conjectures of fundamental importance in harmonic analysis: establishing the dimension-free weak type for certain singular integrals; confirming the exact value of Grothendieck's constant; and solving the Erdos--Falconer conjecture on the sphere. A parallel, but closely related development is building a unified weighted theory of important dyadic operators: maximal functions, square functions, and martingale transforms. What unites all parts of the project is how each problem gives rise to a series of correctly scaling integral estimates. The techniques proposed to obtain those estimates mix the classical arsenal with the Bellman function method. The latter technique, given an especially prominent role in the project, combines optimal control, calculus of variations, non-linear partial differential equations, and differential geometry to establish sharp inequalities.
Two aspects of the project are equally important: solving major open problems and method development. Since the open questions being addressed have proved resistant to attack by traditional theoretical tools, the emphasis is on novel methods that connect several areas of modern mathematics and also borrow from related fields, such as control theory. Each question considered thus has several dimensions: analytical, partial-differential, and geometrical. The structure of the project is incremental and each new partial result should enhance our understanding of the deep connections among these areas. It is expected that the methodology employed will yield a large body of teachable cutting-edge material, some of which may soon be entering graduate curricula and helping bring new researchers into the field.
When studying the behavior of a transformative device (operator) acting within or between classes of signals (functions), one uses a measuring device (functional) to estimate the size of the output in terms of fixed characteristics of the input. This usually entails finding good bounds on a certain integral or sum and, while some bounds may be obtained by classical means, it is often of essence to find the best possible (sharp) bounds. These are typically hard to come by; in many cases of interest they are still unknown. The relatively novel technique at the center of this project, the method of the Bellman function, recasts the problem of estimating an integral or sum functional as that of finding a certain function with subtle size and convexity properties. That function often arises as a solution of a non-linear partial differential equation on a problem-specific domain, in which case its graph forms a surface with minimal allowable curvature. Once in hand, the Bellman function reveals not only the sharp bound sought, but also the extremal configuration of the signal that produces that bound. Motivated by several major open problems in harmonic analysis, the project develops the method in depth and explains how to find Bellman functions in many important settings. A particular focus is on BMO (the class of functions of bounded mean oscillation), and the results obtained lay the foundation of a general Bellman theory of integral functionals on BMO. This theory can be naturally extended to many other function classes with similar geometry. Furthermore, inverse Bellman functions are used to obtain sharp constants in certain cases where direct Bellman dynamics is ill-posed. The project also develops Bellman analysis for classes defined on alpha-trees. This allows one to obtain results sharp in dimension, by replacing it with a flexible parameter that can be adjusted in inductive arguments. In the direction of sharp estimates for sums, a general family of Carleson sequences related to A_2 weights is precisely described in terms of its Carleson norms, generalizing and unifying many earlier results. Also studied are the action of the dyadic maximal operator between weight classes and new exponential estimates for the dyadic square function. By its nature, the project connects several areas of mathematics: harmonic analysis, partial differential equations, variational calculus, and differential geometry. It has also produced a large body of teachable material that has already served as a foundation for a ten-week seminar seminar at the University of Cincinnati, with significant graduate student participation. In addition, this material has been prominently featured at two international Bellman-function schools: one in France in 2012 and one in Sweden in 2013.