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

抽象图形 研究人员建议继续研究领域的多线性奇异积分算子和偏微分方程与哈代空间技术。他的技术包括补偿紧凑性方法，原子分解，并估计奇异积分算子。研究者还提出了在欧氏空间上获得极大算子的尖锐估计。 特别是，他感兴趣的是计算运营商规范的哈迪-Littlewood极大函数对不同的勒贝格空间。 这项建议有两个部分。在第一部分的调查计划使用数学分析技术，（特别是谐波分析），以解决问题的微分方程自然出现在研究动态的流体或在研究运动的波。研究人员认为，他提出的方法将为这些领域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，他建议更好地定量理解平均运算。函数的平均是一个重要而有用的操作，因为它通过平滑其奇点来正则化给定的数据集。 调查人员建议研究如何急剧的某些数量属性的功能转移到他们的平均值。

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.