Mathematical Sciences: Problems on Maximal Multiplier Transforms of Weak Type

小行星9400618 这个建议描述了一个继续的合作工作，在勒贝格空间和最大运营商的乘法器在局部紧交换群。这项工作导致无量纲估计径向乘数，如在欧几里得空间的Riesz变换。它也继续早先的工作在转移理论，这一次与科夫曼和韦斯乘数定理的半团体的运营商。与S. Montgomery-Smith关于布朗运动应用的研究在鞅论中极大函数和平方函数的处理以及Banach空间值函数在具有序对偶群的群上的调和分析中得到了新的结果，并显示出统一性。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。 经典分析通常是指对这个交换群上的函数和算子的分析。这个项目涉及研究类似物的经典分析运营商和估计的背景下，一个抽象的交换群。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. ***