Mathematical Sciences: Research in Classical Harmonic Analysis and Applications to Partial Differential Equations

作者拟在Hardy空间技术下进行多线性奇异积分算子和偏微分方程的研究。他的技术包括补偿紧性方法、原子分解和奇异积分算子的估计。研究者还提出了欧几里得空间上的极大算子的尖锐估计。他特别对计算不同勒贝格空间上Hardy-Littlewood极大函数的算子范数感兴趣。这个建议有两个部分。在第一部分中，研究者计划使用数学分析技术（特别是谐波分析）来解决在流体动力学研究或波浪运动研究中自然出现的微分方程问题。研究者认为，他提出的方法将给这些领域中出现的一些问题提供新的见解，并将加强数学与应用的联系。在第二部分中，他建议对平均操作进行更好的定量理解。函数的平均是一项重要而有用的操作，因为它通过平滑其奇点来使一组给定的数据规范化。研究者建议研究函数的某些定量性质如何迅速地转移到它们的平均值上。

Abstract Grafakos The investigator proposes to pursue research in the area of multilinear singular integral operators and partial differential equations in connection with Hardy space techniques. His techniques include compensated compactness methods, atomic decomposition, and estimates for singular integral operators. The investigator also proposes to obtain sharp estimates for maximal operators on Euclidean spaces. In particular, he is interested in computing the operator norm of the Hardy-Littlewood maximal function on different Lebesgue spaces. This proposal has two parts. In the first part the investigator plans to use mathematical analysis techniques, (in particular harmonic analysis), to solve problems in differential equations which naturally arise in the study of dynamics of fluids or in the study of motions of waves. The researcher believes that the methods that he proposes to use will give new insight to some problems that arise in these areas and will strengthen the connection of mathematics with applications. In the second part, he proposes to work on better quantitative understanding of averaging operations. The averaging of a function is an important and useful operation since it regularizes a given set of data by smoothing out its singularities. The investigator proposes to study how sharply are certain quantitative properties of functions transferred to their averages.