Collaborative Research: Bellman function, Harmonic Analysis and Operator Theory

摘要（Volberg/Nazarov/Treil：0758552/0800243/0800876）所提出的研究是在几何测量理论和谐波分析的交叉点。几何测度理论的主要目标是在看似非结构化的分形模式中找到结构。经典的调和分析研究波的传播，而奇异积分算子的研究是现代方法的重要组成部分。我们的持续研究，以及其他几组数学家在美国和国外的工作，已经证明，新的知识可以通过探索这两个领域之间的相互作用。由于我们的建议，我们希望解决几个重要的问题，几何测度理论以及调和分析。模式识别（即，几何测度理论中的问题）将通过使用源于调和分析的方法来推进，反之亦然。我们还希望开发新的方法来研究模式和波。预计这些新开发的技术将对工程和计算机科学的邻近领域产生影响，如图像处理和数据压缩。

Abstract (Volberg/Nazarov/Treil: 0758552/0800243/0800876)The proposed research is at the intersection of Geometric Measure Theory and Harmonic Analysis. The main objective of Geometric Measure Theory is to find structures in seemingly unstructured, fractal-like patterns. The classical Harmonic Analysis studies wave propagation, and investigation of singular integral operators is a crucial part of the modern approach. Our continuing research, as well as the work of several other groups of mathematicians in the US and abroad, has demonstrated that new knowledge can be obtained by exploring the interaction between these two areas. As a result of our proposal we expect to solve several important problems in Geometric Measure Theory as well as in Harmonic Analysis. The pattern recognition (i.e., problems in Geometric Measure Theory) would be advanced by using methods originating in Harmonic Analysis, and vice versa. We also expect to develop new methods to study both patterns and waves. It is expected that these newly developed techniques will have impact to adjacent areas of engineering and computer sciences such as image processing and data compression.