Zeta Functions and Sharp Fractional Integral Inequaities

摘要提案：DMS-9622891 PI：Morpurgo 本研究的主要目的是：1.显式计算谱不变量所产生的zeta函数的自然伪微分算子，在共形变化的度量; 2。分析这种不变量的空间，如n维球，环面，球和壳，通过已知的或新的尖锐索伯列夫或分数积分不等式。 Morpurgo建议对紧致流形（有边界或无边界）上的一大类共形协变伪微分算子和一些非共形协变算子（如维度大于2的普通拉普拉斯算子）进行研究。计算将通过研究者开发的变分或直接新方法来实现。空间上的不变量的显式形式，如球，环面，球和壳，然后将与已知的尖锐的不平等，或将建议可能的新的尖锐的不平等，由研究人员使用对称化和端点微分参数。 理解微分算子的谱如何随着几何的变化而变化是数学和物理学中的一个基本问题。研究结果将为共形协变算子的谱控制提供具体而直接的方法，反之亦然。这种算子在许多物理现象的研究中自然出现，如电磁学、统计力学和随机表面理论。

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.