Mathematical Sciences: Problems on Maximal Multiplier Transforms of Weak Type

9400618阿斯马尔这一建议描述了在勒贝格空间上的乘子和局部紧交换群上的极大算子上的合作工作的继续。这项工作导致了欧氏空间上的径向乘子的无量纲估计，例如Riesz变换。它还继续了早期在迁移理论方面的工作，这一次是关于半算子群的Coifman和Weiss乘子定理。与S.Montgomery-Smith在布朗运动应用方面的合作正在产生新的结果，并在处理鞅理论中的极大值函数和平方函数以及具有序对偶群的Banach空间值函数的调和分析方面显示出统一性。N维欧氏空间中的向量可以相加，这样欧氏空间就是一个交换群。经典分析通常指的是对这个交换群上的函数和算子的分析。这个项目涉及到在抽象交换群的背景下研究经典分析算子和估计的类似。这种抽象的观点导致了大范围分析的统一。***

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. ***