Mathematical Sciences: Problems on Maximal Multiplier Transforms of Weak Type

9400618 Asmar 该提案描述了勒贝格空间上的乘法器和局部紧凑阿贝尔群上的最大算子的协作工作的延续。这项工作导致了径向乘数的无量纲估计，例如欧几里得空间上的 Riesz 变换。它还继续了转移理论的早期工作，这次与半算子群的 Coifman 和 Weiss 乘子定理相关。与 S. Montgomery-Smith 在布朗运动应用方面的联合工作正在产生新的成果，并显示了鞅理论中极大函数和平方函数的处理以及具有有序对偶群的 Banach 空间值函数的调和分析的统一性。 n 维欧几里得空间中的向量可以相加，这样欧几里得空间就是一个交换群。 经典分析通常是指对这个交换群上的函数和算子的分析。该项目涉及研究经典分析算子的类似物以及抽象交换群背景下的估计。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. ***