Singular Integrals and Maximal Functions

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Abstract for Proposal 0098757I plan to study analytical estimates for certain integral operators defined on functions on the Euclidean space of dimension greater than or equal to two. In these operators the integration is over surfaces of positive codimension, and we seek estimates reflecting curvature properties of the surface. Suppose for example, for each point, P, inthe Euclidean space we have a one dimensional curve emanating from P. From a given function, f, we form a new function Mf, called the maximal function, whose value at the point P is the supremum of the averages of f over the curve emanating from P. This then defines a transformation from functions on the Euclidean space to functions onthe Euclidean space. We want to know for what curves and what values of p this transformation is bounded on the Lebesgue space of functions with integrable pth power. Positive results here imply variants of Lebesgue's theorem on the differentiation of the integral. Namely ifthe transformation from a function f to Mf is bounded on one of these Lebesgue spaces, then for every function f in that Lebesgue space and almost every point P, the value f(P) may be recovered as a limit of averages of f over small portions of the curve through P. I am alsointerested in discrete analogues of these operators in which integration is replaced by sums over discrete sets of points.A basic problem for over a hundred years of the branch of mathematics known as Classical Analysis is that of recovering a function from averages of that function. This problem has been intimately connected with that of approximating an arbitrary function by a combination ofsimpler functions which in turn has been one of the main ways mathematics is applied to real world problems. I plan to study the problem of recovering functions on Euclidean space of at least two dimensions, from averages over small pieces of one dimensional curves. For a continuous function this is an easy question, but for wildly discontinuous functions it is a subtle problem depending on curvatureproperties of the curves. I also plan to study relatedtransformations and discrete analogues of these transformations where the averaging process is over discrete sets of points.