Quantization on Lie groups

该研究将集中于非对易量子化理论的发展，并进一步应用于相空间分析、时频分析和偏微分方程组理论等领域。在拟议的分析中，有许多重要的例子激发了巨大的需求。幂零李群在一般调和分析和流形上涉及偏微分算子的问题中的重要性早已被认识到。这些问题可以追溯到著名的Hörmander平方和定理，Rothschild-Stein提升定理，齐次Lie群上Hardy空间的Folland-Stein工作，以及Heisenberg流形上的Beals-Greiner演算。我们感兴趣的是开发一种新的方法来研究幂零和其他非对易环境中的伪微分算子，以便在一般理论中取得进展，但要记住群和偏微分方程组的所有特别重要的激励例子，以及在时频分析中的应用。非对易环境下算子的量子化问题在偏微分方程界由来已久，而且是出了名的困难。这个建议的目的之一是建立在紧李群的量子化理论的最新进展以及最近关于Heisenberg群上符号的量子化的工作的基础上，并进一步应用于Heisenberg流形上的问题，最著名的是次椭圆估计和指数定理的问题。从这个角度来看，最近引入的诸如差分算符等概念将流形上的一般量子化与Calderón-Zygmund算子的Coifman-Weiss理论联系起来，新兴的符号量子化技术为解决这些问题提供了一种新的尝试。然而，要执行这个程序，肯定有许多有趣的障碍需要克服，例如在局部紧环境下使用适当的C*-代数语言进行傅立叶分析，在非分层环境中或在一般阶的微分算子的情况下，考虑群结构的适当的Sobolv空间的开发，从而找到在非对易环境中连接多个分析领域的方法。因此，在构造了令人满意的算子的符号演算之后，我们计划将其应用于偏微分方程组理论中的问题，这是此类分析的最重要的目标之一。这将包括发展偏微分方程传播子的符号表达式，允许得到它们的必要估计(能量、Strichartz、光滑化)，建立关于群结构的椭圆或亚椭圆算子的全局下界，以及通过群作用来解决球面上向量场的长期存在的全局可解性问题(Greenfield-Wallach和Katok猜想)。这是一项重要的、具有挑战性的、及时的研究，对非对易算子分析和偏微分方程理论以及它们与其他领域和应用的关系具有深刻的影响。

The proposed research will concentrate on the development of the non-commutative quantization theory with further applications to areas such as phase space analysis, time-frequency analysis, and the theory of partial differential equations. There are many important examples motivating the great need in the proposed analysis. The importance of nilpotent Lie groups has been realised since a long time in general harmonic analysis as well as in problems involving partial differential operators on manifolds. Such questions go back to the celebrated Hörmander's sum of the square theorem, to the Rothschild-Stein lifting theorems, the Folland-Stein work on the Hardy spaces on homogeneous Lie groups, and Beals-Greiner calculus on the Heisenberg manifolds. We are interested in developing a new approach to pseudo-differential operators in the nilpotent and other non-commutative settings, to make advances in a general theory, but keeping in mind all the particular important motivating examples of groups and of PDEs, with applications to the time-frequency analysis. The problem of the quantization of operators in the non-commutative setting is long-standing and notoriously difficult in the area of partial differential equations. One of the aims of this proposal is to build up on the recent advances in the quantization theory on compact Lie groups as well as on recent works on the quantization of symbols on the Heisenberg group with further applications to problems on Heisenberg manifolds, most notoriously ones of the subelliptic estimates and of the index theorems. From this point of view the recently introduced concepts such as those of difference operators linking the general quantization on manifolds to the Coifman-Weiss theory of Calderón-Zygmund operators, the emerging techniques of symbolic quantization provide for a possibility to making a new attempt at tackling these problems. However, there are certainly many interesting obstacles one needs to overcome to carry out this program, e.g. using an appropriate C*-algebra language for the Fourier analysis in the locally compact setting, development of appropriate Sobolev spaces taking into account the group structure in the non-stratified setting or in the case of differential operators of general orders, thus finding ways to linking several areas of analysis in the non-commutative setting.Consequently, having constructed the satisfactory symbolic calculus of operators, we plan to apply this to the problems in the theory of partial differential equations, which is one of the most important objectives for such analysis. This will include symbolic expressions for propagators of evolution partial differential equations allowing for deriving necessary estimates for them (energy, Strichartz, smoothing), establishing global lower bounds for operators elliptic or hypoelliptic with respect to the group structure, as well as to long-standing global solvability problems for vector fields on spheres through the group action (Greenfield-Wallach and Katok conjectures).This is important, challenging and timely research with deep implications in the theories of non-commutative operator analysis and partial differential equations, as well as their relations to other areas and applications.