Maximal Averages Over Hypersurfaces

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ABSTRACT Iosevich The purpose of this project is to analyze various operators that arise in harmonic analysis and partial differential equations associated with the convex finite type hypersurfaces. The main focus of the project is the study of the maximal averaging operators introduced by E. M. Stein, and the averaging convolution operators originally studied by Littman and Strichartz. Iosevich proposes a set of necessary and sufficient conditions for the Lp boundedness of maximal averaging operators, and a set of necessary and sufficient conditions for the (Lp, Lq) boundedness of the averaging convolution operators. Both sets of conditions are expressed in terms of the integrability of the negative powers of the distance functions to the tangent planes to the hypersurface. Iosevich will also work on the problem of Lp boundedness of the maximal Bochner-Riesz means associated with the smooth convex bodies in Rn. It has been conjectured that these operators should have the same mapping properties as the standard maximal Bochner-Riesz means associated with the ball. The study of the maximal averaging operators and other similar operators in harmonic analysis is partially motivated by the following interesting question: How close can we come to recovering a set of data from the various kinds of averages of that data? The question is of potential practical value since scientists are often called upon to make predictions based on average information. For example, meteorologists make predictions about the rainfall in the particular location based on the average rainfall in the years past in the nearby towns. Seismologists make earthquake predictions based on the pattern of shocks in the surrounding area. The tradeoff involved in the study of these phenomena is, roughly speaking, the following. If the data is very precise, then it can, generally speaking, be recovered from any kind of a reasonable average. If the data is less pre cise, then we have to make sure that the averaging process compensates the deficiencies of the data. The main thrust of this project is to study the averaging phenomenon when the data is given by a certain kind of a mathematical function, and the average is taken over a curved surface.