Mathematical Sciences: Harmonic Analysis, Sobolev Inequalities and Geometry

有三个主题突出了这一数学研究。第一种方法结合了谐波分析和概率论的思想。它们涉及无穷级数的哈尔函数——在区间上定义的正交阶跃函数。任何交替和的幂范数总是由原范数的一个固定倍数支配。虽然最尖锐的乘数是已知的，但使范数最大化或最小化的交替和并不容易得到。在这个项目中，我们将根据原始级数的系数来确定符号变化的最优选择。第二条研究线将集中在球面上拉普拉斯算子的指数倍数上。这种算子的行列式的对数包含一个非负能量项，只有当指数是基本共形方程中的势时，该项才会消失。这个结果在一个较低的维度上有意义，因为它变成了经典的Lebedev-Milin不等式，而这个不等式又衍生自Szego定理。我们将努力在更高的维度中扭转这个论点；从塞戈定理的一个版本到能量不等式。这项研究的基础是关于拉普拉斯谱的基本问题。更有几何趣味的工作将继续讨论球面上定义的哪些函数可以是共形相关度量的曲率函数的问题。根据函数的临界点，已知充分的拓扑条件。根据函数的解析性质寻求充要条件。

Three main themes highlight this mathematical research. The first combines ideas of harmonic analysis and probability theory. They involve infinite series of Haar functions - orthogonal step functions defined on an interval. The power norm of any alternating sum is always dominated by a fixed multiple of the original norm. Although the sharpest multiplier is known, the alternating sums which maximize or minimize the norm are not easily obtained. In this project, work will be done in determining optimal choices of sign change based on the coefficients of the original series. A second line of investigation will focus on exponential multiples of the Laplace operator on the surface of a sphere. The logarithm of the determinant of such an operator contains a nonnegative energy term which vanishes only when the exponential is the potential in the basic conformality equation. This result has implications in one lower dimension in that it becomes the classical Lebedev-Milin inequality which, in turn, derives from Szego's theorem. Efforts will be made in reversing the argument in higher dimensions; from a version of the Szego theorem to the energy inequality. Underlying this research are basic questions concerning the spectrum of the Laplacian. Work of a more geometric flavor will continue on the question of which functions defined on a sphere can be the curvature functions of conformally related metrics. Sufficient topological conditions are known depending on the critical points of the function. Necessary and sufficient conditions based on analytic properties of the function will be sought.