Mathematical Sciences: Problems on Maximal Multiplier Transforms of Weak Type

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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. ***