Zeta Functions and Sharp Fractional Integral Inequaities

提案：DMS-9622891 PI: Morpurgo提出的主要研究目标是：1。由自然伪微分算子的zeta函数引起的谱不变量的显式计算，根据度量的保角变化2. 利用已知的或新的尖锐Sobolev或分数阶积分不等式，分析诸如n维球面、环面、球和壳等空间上的不变量。Morpurgo提出对紧流形（有边界或无边界）上的一大类共形协变伪微分算子，以及一些非共形协变算子（如大于2维的普通拉普拉斯算子）进行这一研究方案。计算将通过研究者开发的变分或直接新方法来实现。空间上不变量的显式形式，如球体，环面，球和壳，然后将与已知的尖锐不等式相关，或者将提出可能的新的尖锐不等式，由研究者使用对称和端点微分论证推导。理解微分算子的谱如何随着底层几何的变化而变化是数学和物理中的一个基本问题。研究结果将提供具体和直接的方法来根据曲面的度量结构来控制共形协变算子的谱，反之亦然。这样的算子自然出现在许多物理现象的研究中，如电磁学、统计力学和随机表面理论等领域。

ABSTRACT Proposal: DMS-9622891 PI: Morpurgo The main goals of the proposed research are: 1. Explicit computations of spectral invariants arising from the zeta function of natural pseudodifferential operators, in terms of conformal changes of metrics; 2. Analysis of such invariants on spaces such as n-dimensional spheres, tori, balls and shells, via either known or new sharp Sobolev or fractional integral inequalities. Morpurgo proposes to carry out this research program for a large class of conformally covariant pseudodifferential operators on a compact manifold (with or without boundary), and for some non-conformally covariant operators, such as the ordinary Laplacian in dimensions greater than 2. The computation will be effected via either variational or direct new methods developed by the investigator. The explicit form of the invariants on spaces like spheres, tori, balls and shells will then be related to either known sharp inequalities, or will suggest possible new sharp inequalities, to be derived by the investigator using symmetrization and end-point differentiation arguments. Understanding how the spectrum of differential operators change as the underlying geometry is varied is a fundamental problem in mathematics and physics. The results produced in the research will provide concrete and direct ways to control the spectrum of conformally covariant operators in terms of the metric structure of surface, and vice versa. Such operators arise naturally in the study of many physical phenomena, in fields such as electromagnetism, statistical mechanics, and the theory of random surfaces.