The Role of Gaussian Curvature in Harmonic Analysis and Related Areas

文摘：作者建议研究调和分析及相关领域中的一系列问题，其中高斯曲率起着重要作用。高斯曲率是以超曲面上的点为单位法线的高斯映射的微分的行列式。更具体地说，作者建议研究以下三个基本问题：曲面平均的正则性，凸域中格点的分布，以及欧氏空间中域的正交指数基的存在性。在曲面上的平均的背景下，我们建议根据自然的和可计算的几何准则，找到这些平均的勒贝格空间有界的一组充要条件。在凸域格点的背景下，我们建议根据边界的自然几何性质，计算膨胀的凸体的体积与被困在其中的格点数目之间的差异的急剧增长率。在正交指数基的背景下，我们建议对Fuglede猜想的证明取得进展。Fuglede猜想说，一个域有正交指数基当且仅当它可以用该域的不交平移平铺欧几里得空间。组合和数论方法有望发挥重要作用。调和分析中最大平均算子和其他类似算子的研究部分是由以下有趣的问题引起的：我们能从数据的各种平均值中恢复一组数据有多近？这个问题具有潜在的实用价值，因为科学家经常被要求根据平均信息进行预测。例如，气象学家根据附近城镇过去几年的平均降雨量对特定地点的降雨量进行预测，地震学家根据周围地区的地震模式进行地震预测。粗略地说，研究这些现象所涉及的权衡如下。如果数据非常精确，那么一般来说，它可以从任何一种合理的平均值中恢复出来。如果数据不那么精确，那么我们必须确保平均过程弥补了数据的不足。本项目的主要目的是研究由某种数学函数给出的数据在曲面上取平均值时的平均现象。研究凸域中格点的分布和相关的差异函数的动机是希望用更容易计算的连续信息来近似差异信息，例如平面上的整点，在这种情况下是面积。最后，用三角函数逼近函数是一个重要的实际问题，这类逼近在物理、工程和许多其他科学技术领域有着广泛的应用，因此对正交指数基的研究受到了启发。

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.