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

该项目致力于解决调和分析中三个长期存在的基本猜想的部分或全部问题：建立某些奇异积分的无量纲弱型；确定Grothendieck常数的精确值；以及求解球面上的Erdos-Falconer猜想。一个平行但密切相关的发展是建立重要并元算子的统一加权理论：极大函数、平方函数和鞅变换。将项目的所有部分联系在一起的是每个问题如何产生一系列正确的定标积分估计。为获得这些估计而提出的技术混合了经典的武器库和贝尔曼函数方法。后一种技术在该项目中的作用尤为突出，它结合了最优控制、变分法、非线性偏微分方程和微分几何来建立尖锐的不等式。该项目的两个方面同等重要：解决重大开放问题和方法开发。由于正在解决的开放问题已被证明抵抗了传统理论工具的攻击，因此重点是连接现代数学的几个领域并借鉴相关领域(如控制论)的新方法。这样考虑的每个问题都有几个维度：解析性、偏微分性和几何性。该项目的结构是渐进的，每一项新的部分成果都应加强我们对这些领域之间的深刻联系的理解。预计所采用的方法将产生大量可教授的尖端材料，其中一些可能很快就会进入研究生课程，并帮助将新的研究人员带入该领域。

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.