The Role of Gaussian Curvature in Harmonic Analysis and Related Areas

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ABSTRACT:The author proposes to study a set of problems in harmonic analysis andrelated areas where the Gaussian curvature, the determinant of thedifferential of the Gauss map taking the point on a hypersurface to theunit normal at that point, plays an important role. More specifically,the author proposes to study the following three basic problems:regularity of averages over surfaces, distribution of lattice points inconvex domains, and the existence of orthogonal exponential bases fordomains in Euclidean space. In the context of averages over surfaces, wepropose to find a set of necessary and sufficient conditions for theLebesgue space boundedness of these averages in terms of natural andeasily computable geometric criteria. In the context of lattice pointsin convex domains, we propose to compute a sharp rate of growth for thediscrepancy between the volume of a dilated convex body and the numberof lattice points trapped inside, again in terms of natural geometricproperties of the boundary. In the context of orthogonal exponentialbases, we propose to make progress towards the proof of the FugledeConjecture, which says that a domain has orthogonal exponential basis ifand only if it is possible to tile Euclidean space with disjointtranslates of this domain. Combinatorial and number theoretic methodsare expected to play an important role.The study of the maximal averaging operators and other similar operatorsin harmonic analysis is partially motivated by the following interestingquestion: How close can we come to recovering a set of data from thevarious kinds of averages of that data? The question is of potentialpractical value since scientists are often called upon to makepredictions based on average information. For example, meteorologistsmake predictions about the rainfall in a particular location based onthe average rainfall in years past in nearby towns.Seismologists make earthquake predictions based on the pattern of shocksin the surrounding area. The tradeoff involved in the study of thesephenomena is, roughly speaking, the following. If the data is veryprecise, then it can, generally speaking, be recovered from any kind ofa reasonable average. If the data is less precise, then we have to makesure that the averaging process compensates for the deficiencies of thedata. The main thrust of this project is to study the averagingphenomenon when the data is given by a certain kind of a mathematicalfunction, and the average is taken over a curved surface. The study ofthe distribution of lattice points in convex domains and the associateddiscrepancy function is motivated by the desire to approximate discreteinformation, for example integer points in the plane, by more easilycomputable continuous information, in this case the area. Finally, thestudy of orthogonal exponential bases is motivated by an importantpractical problem of approximating functions by trigonometric functions.These types of approximations have numerous applications in physics,engineering, and many other areas of science and technology.