Weighted, non-local, and product-type Bellman estimates in Harmonic Analysis

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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.