Spectral multipliers on nilpotent Lie groups and homogeneous spaces

对于L^2-空间上的强交换自伴算子系统，利用谱积分定义了一个联合泛函演算，其中有界Borel函数对应于L^2-有界算子.这类算子的L^p-有界性（对于p的其他值）用定义函数（称为谱乘子）来刻画要困难得多。调和分析的一些问题和结果都属于这个框架，其中涉及各种技术，如奇异积分算子的Calderon-Zygmund理论，并在偏微分方程的研究中有应用。本项目主要研究幂零李群上群不变微分算子系统的情形。虽然有相当普遍的定理给出了乘子光滑性的L^p有界性的充分条件，但只有在少数情况下才能证明尖锐的结果，这需要详细了解底层的代数结构，并强调欧几里得几何结构和次黎曼几何结构之间的非平凡相互作用。最近发展了一种新的方法来处理2阶群上齐次拉普拉斯算子的情形。这个项目的目的是扩展新的技术，并获得尖锐的L^p-有界性的结果，更广泛的群体和运营商，以及系统的交换运营商。

For a system of strongly commuting self-adjoint operators on some L^2-space, a joint functional calculus is defined via spectral integration, where bounded Borel functions correspond to L^2-bounded operators. L^p-boundedness (for other values of p) of such operators is much more difficult to characterize in terms of the defining functions, called spectral multipliers. Several problems and results of harmonic analysis fall into this frame, which involve various techniques such as the Calderon-Zygmund theory of singular integral operators, and have applications in the study of partial differential equations. This project is mainly concerned with the case of a system of group-invariant differential operators on a nilpotent Lie group. Although quite general theorems giving sufficient conditions for the L^p-boundedness in terms of smoothness of the multiplier are available, only in few cases sharp results have been proved, requiring a detailed knowledge of the underlying algebraic structure and highlighting a nontrivial interaction between Euclidean and sub-Riemannian geometric structures. Recently a new technique has been developed to deal with the case of a homogeneous sublaplacian on a 2-step group. The aim of this project is to extend the new technique and to obtain sharp L^p-boundedness results for wider classes of groups and operators, and for systems of commuting operators.