Mathematical Sciences: Problems on Maximal Multiplier Transforms of Weak Type

这个提议描述了Lebesgue空间上乘数和局部紧阿贝尔群上的极大算子的协同工作的延续。这项工作导致了径向乘法器的无量纲估计，如欧几里得空间上的Riesz变换。它还继续了早先在迁移理论方面的工作，这一次是与半算子群的Coifman和Weiss乘数定理有关。与S. Montgomery-Smith合作研究布朗运动的应用，得到了新的结果，显示了鞅理论中对极大函数和平方函数处理的统一性，以及Banach空间值函数在有序对偶群上的调和分析。n维欧几里德空间中的向量可以相加，因此欧几里德空间是一个交换群。经典分析通常是指对这个交换群上的函数和算子的分析。本项目涉及在抽象交换群的背景下研究经典分析算子和估计的类似物。这种抽象的观点导致了大面积分析的统一。＊＊＊

9400618 Asmar This proposal describes a continuation of collaborative work in multipliers on Lebesgue spaces and maximal operators over a locally compact abelian group. The work leads to dimension-free estimates for radial multipliers such as the Riesz transforms on Euclidean space. It also continues earlier work in transference theory, this time in connection with the Coifman and Weiss Multiplier Theorem for semi-groups of operators. The joint work with S. Montgomery-Smith on the applications of Brownian motion is yielding new results and showing a unity in the treatment of maximal and square functions in martingale theory, and harmonic analysis of Banach space valued functions on groups with ordered dual groups. Vectors in n-dimensional Euclidean space can be added and in this way Euclidean space is a commutative group. Classical analysis usually refers to analysis of functions and operators on this commutative group. This project involves studying analogues of classical analysis operators and estimates in the context of an abstract commutative group. This abstract point of view leads to a unification of large areas of analysis. ***