Singular Integrals and Maximal Functions

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Proposal: DMS-973647Principal Investigator: Stephen WaingerAbstract: Wainger plans to study maximal and singular averages of functions defined on n-dimensional space, where n is greater than or equal to 2. In contrast to the theory of Calderon and Zygmund, the integration in these averages is to take place over curves or surfaces of positive codimension. Wainger is interested in L-p bounds for these operators that relate to curvature properties of the curves and surfaces. He also plans to study discrete analogues of these questions, in which integration is replaced by a summation over a discrete set of points.An important problem in mathematics in the last 250 years has been that of approximating arbitrary functions by combinations of simpler functions. This problem arose in the study of vibrating strings in the 18th century and in the study of heat flow in the 19th century. More recently this problem has been the focus of considerable activity in conjunction with the electronic transmission of data. This approximation problem has led in modern times to the problem of finding estimates for the sizes of certain "averages" of functions in terms of the size of the functions. Wainger's proposed research involves the averaging of functions of n variables, with n greater than or equal to 2, now over surfaces or curves in n-dimensional space rather than all of n-dimensional space. He then seeks estimates for these averages in terms of geometric properties of the surfaces or curves, as well as of the functions themselves.