Mathematical Sciences: Geometric Harmonic Analysis and Spectral Theory

这项工作的基础是几何分析和调和分析之间存在的强大联系。后一个主题涉及将数学函数分解成更简单的片断，以及从这些片断重构(合成)函数。通常，这些片断是作为定义在曲面或流形上的拉普拉斯算子等微分算子的本征函数自然产生的。然而，算子的本征函数出现在调和分析思想刚刚开始渗透的许多环境中。这项研究的目的是促进这一方向的进一步发展。第三步或更高阶的单连通李幂零群有一个次椭圆拉普拉斯量，它可以表示为生成元的平方和。将在确定该算子的本征函数方面做工作。对于第二步群，如Heisenberg群，用Hermite函数来描述本征函数。目前的工作应该会产生新的特殊功能类别。其他工作将研究分数Hausdorff维集。这些集合长期以来一直在调和分析中发挥作用，现在正在数学的许多领域中变得突出起来。这个项目的一个目标将是研究最近发现的傅立叶变换均方积分和原始函数关于阿尔法维度量的相同积分之间的更紧密的不等式。特别是，工作将集中在通过考虑乘以该度量的函数的傅里叶变换的渐近行为来确定这种度量的支撑性的什么分形性。一条新的研究路线将采用贝西科维奇在本世纪初发展起来的概周期函数的经典理论。该理论的缺陷之一是，当为这些函数的空间定义范数时，它会产生很大的零空间。这项工作将寻找在空间上产生度量的新方法，发展分布观点，并推导出空间上的广义调和分析的有用理论，特别是强调傅立叶变换的展开。

The basis for this work is the strong connections which exist between geometric and harmonic analysis. The latter subject concerns the decomposition of mathematical functions into simpler pieces and the reconstruction (synthesis) of functions from these pieces. Often the pieces arise naturally as eigenfunctions of differential operators such as the Laplace operator defined on a surface or manifold. However, eigenfunctions of operators occur in many contexts where the ideas of harmonic analysis are just beginning to penetrate. The object of this research is to foster further development in this direction. A simply-connected Lie nilpotent group of step 3 or higher has a sub-elliptic Laplacian representable as the sum of the squares of the generators. Work will be done in determining the eigenfunctions of this operator. For step 2 groups such as the Heisenberg group, the eigenfunctions are described in terms of Hermite functions. The present work should lead to new classes of special functions. Other work will study sets of fractional Hausdorff dimension. These sets which have long played a role in harmonic analysis are now coming into prominence in many areas of mathematics. One goal of this project will be to examine more closely recently discovered inequalities between the mean-square integral of Fourier transforms and the same integral of the originating function taken with respect to alpha-dimensional measure. In particular, work will concentrate on determining what fractal properties of the support of such a measure can be obtained by considering the asymptotic behavior of Fourier transforms of functions multiplied against the measure. A new line of investigation will take up the now classical theory of almost periodic functions developed early in the century by Besicovitch. One of the drawbacks to the theory was the fact that when a norm was defined for the space of such functions, it produced a large null space. This work will look for new ways of producing metrics on the space, developing a distributional point of view and deriving a useful theory of generalized harmonic analysis on the space, with special emphasis placed on the expansion of the Fourier transform.