Singular Integrals and Maximal Functions

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

ABSTRACT Professor Wainger expects the primary focus of his research to be on l(p) estimates for discrete analogues of a family of continuous operators. In the continuous situation one is given for each point, x, in n-dimensional space a variety, V (x), of positive codimension passing through x. Given a function, f , on n-dimensional space one considers maximal or singular averages of f where the average of x is over the variety V (x). We then seek L(p) estimates for these averages. In the discrete analogue we consider functions defined over the lattice points, m, in n-dimensional space, and the average is over an arithmetic set containing m. In the continuous problem alluded to above, most of the work assumes the variety V (x) has finite order of contact with its tangent plane. Professor Wainger also plans to continue his investigation into the case that V (x) has infinite order of contact with its tangent plane. Professor Wainger plans to study averages of functions defined on lattice points in n-dimensional space. (Lattice points are points with integral coordinates.) For each lattice point, m, one considers averages over an arithmetic subset containing m. The goal is to relate analytical properties of these averages to arithmetic properties of these subsets. Results of this type have applications to problems of Ergodic theory. A corresponding contin uous problem has received a great deal of attention in the last forty years. In this problem one averages functions defined on n-dimensional space over submanifolds of n-dimensional space. The goal is to relate analytical esti mates for these averages to geometric properties of the submanifold. This work largely assumes that there is only finite order of contact between the submanifold and its tangent plane. Professor Wainger plans to continue his study of the situation in which the submanifold has infinite order of contact with its tangent plane.