Mathematical Sciences: Harmonic Analysis, Sobolev Inequalities and Geometry

三个主要主题突出了这项数学研究。第一种方法结合了调和分析和概率论的思想。它们涉及无穷级数的Haar函数--定义在一个区间上的正交阶梯函数。任意交错和的幂范数总是由原范数的固定倍数所支配。虽然最尖锐的乘子是已知的，但使范数最大化或最小化的交错和并不容易得到。在这个项目中，将根据原始序列的系数来确定符号变化的最佳选择。第二行的研究将集中在球面上拉普拉斯算子的指数倍数。这种算符的行列式的对数包含一个非负能量项，只有当指数是基本一致方程中的势时，它才会消失。这一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Lebedev-Milin不等式又源于Szego定理。将努力在更高的维度上扭转这一论点；从Szego定理的一个版本到能量不平等。这项研究的基础是关于拉普拉斯谱的基本问题。关于定义在球面上的哪些函数可以是共形相关度量的曲率函数的问题，将继续进行更具几何色彩的工作。根据函数的临界点，充分的拓扑条件是已知的。基于函数的解析性质，我们将寻求充分必要条件。

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.