Harmonic Analysis and Affinely Invariant Measures

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AbstractThis project will focus on the relationship of affinely-invariant measures tocertain problems in harmonic analysis. Drury observed that affine arclengthappears to be the natural measure to use when considering averaging and Fourierrestriction operators associated with certain curves. More recently, the PI hasobtained certain uniform results along these lines for curves in two dimensionsand for surfaces in higher dimensions. He intends to continue theseinvestigations and also to consider some questions of an integral-geometricnature. The latter stem from the concept of affine dimension, an analog ofHausdorff dimension which also takes into account a set's curvature.The averaging operators considered in this project belong to a class of smoothingoperators which play an important role in the study of physical processes likewave motion and heat propagation. These operators are also related to the Radontransforms which appear in tomography. Similar operators are important in signalanalysis and communications theory. The Fourier restriction operators consideredhere are part of the theory of Fourier analysis. Their study contributes to thegeneral understanding of the Fourier transform, a tool which is vital toapplications of mathematics ranging from electrical engineering andcommunications theory through fluid dynamics.