Sharp Estimates for Eigenvalues, Integrals, and Sobolev Imbeddings

作者建议研究用积分或本征值定义的泛函的各种极值问题。这些问题包括：关于薛定谔算子有界态的Lieb-Thirring型不等式；关于CR球面的尖端的Moser-Trudinger和Soblev不等式，及其在CR不变算符的行列式中的应用；涉及交叉比的积分的尖端不等式，及其在球面上的Laplian函数的Zeta函数上的应用；圆盘上的对数-Sobolev型的尖端不等式，以及对圆盘上的Dirichlet和Neumann LaPlacian的Zeta函数的应用；无量纲Carleson测度不等式。这项研究将涉及微分几何、调和分析和数学物理中出现的重要问题。很难低估特征值估计在应用科学和理论科学中的重要性。通常的情况是，对于给定的物理结构，例如粒子系统、振膜或弦，人们可以附加某些特征数值，称为“本征值”。这些数字很重要，因为给定结构的随机性通常可以编码成本征值的数值函数，称为“谱泛函”。在极值问题中，人们试图找出同一类型的哪些结构可能具有最高(或最低)可能的谱泛函；通常，它们是表现出最对称几何形状的结构。

The author proposes to work on various extremal problemsfor functionals which are defined in terms of integralsor eigenvalues. These problems include: Lieb-Thirringinequalities for the number bound states of Schrodinger operators; sharp Moser-Trudinger and Sobolev inequalitiesfor the CR sphere, with applications to determinants of CR invariant operators; sharp inequalities for integralsinvolving cross-ratios, with applications to zeta functionsof Laplacians on the sphere; sharp inequalities of log-Sobolevtype on the disk, with applications to zeta functions ofDirichlet and Neumann Laplacians on the disk; dimension-free Carleson measure inequalities. The research willtouch on important questions arising in differentialgeometry, harmonic analysis, and mathematical physics.It is difficult to understate the importance of eigenvalueestimates in both applied and theoretical sciences. It isvery often the case that to given physical structures, suchas systems of particles, vibrating membranes, or strings,one can attach certain characteristic numerical values,called "eigenvalues". These numbers are important, as somerelevant properties of a given structure can be often codedinto numerical functions of the eigenvalues, called"spectral functionals". In an extremal problem one triesto find out which structures of the same type wouldyield the highest (or lowest) possible spectral functionals; very often, they are the ones with exhibit the mostsymmetrical geometry.