Collaborative Research: Bellman function, Harmonic Analysis and Operator Theory

摘要(沃尔伯格/纳扎罗夫/特雷尔：0758552/0800243/0800876)拟议的研究处于几何测度论和调和分析的交叉点上。几何测度论的主要目标是在看似无结构的、类分形的图案中寻找结构。经典的调和分析研究波的传播，而对奇异积分算子的研究是现代分析方法的重要组成部分。我们继续进行的研究，以及美国和国外其他几组数学家的工作表明，通过探索这两个领域之间的相互作用，可以获得新的知识。作为我们的建议的结果，我们期望解决几何测度论和调和分析中的几个重要问题。模式识别(即几何测度论中的问题)可以用起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邻近的工程和计算机科学领域产生影响，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.