Sub-Elliptic Harmonic Analysis

调和分析的许多问题和结果都与欧几里德空间上的拉普拉斯算子有关。拉普拉斯函数出现在许多重要的微分方程中（描述诸如热扩散、波传播或量子动力学等物理现象），对它的研究有助于分析这些方程的解（从而有助于理解所述现象）。特别着重于拉普拉斯泛函演算中算子的有界性与相应谱乘子的平滑性之间的关系。尽管在过去的几十年里取得了一些令人兴奋的突破，但这一领域的许多重要问题，如Bochner-Riesz猜想，仍然没有解决。然而，基本的有界性性质已经被很好地理解，在某种程度上，这些有界性结果的鲁棒版本已经被证明，其中拉普拉斯算子可以用更一般的椭圆算子代替。然而，椭圆性并不总是一个自然的假设。在许多情况下，特别是在存在亚黎曼几何结构的情况下，拉普拉斯式的自然替代不一定是椭圆的，它可能只是亚椭圆的。亚黎曼几何结构和亚椭圆算子在许多数学领域（如复分析和CR几何、非交换李群）中普遍存在，并且在应用领域（如控制理论和机器人技术以及神经生物学）中具有越来越重要的意义。在这种情况下，即使是关于亚椭圆算子的函数有界性的基本问题也远远没有得到解决，已知的结果利用了来自不同数学领域（微分几何、代数和表示理论、泛函和调和分析）的混合技术。提出的研究旨在通过研究特别重要的例子和开发更健壮的技术，在理解亚椭圆算子函数的有界性及其与底层几何的关系方面取得实质性进展。非欧几里得调和分析的长期开放问题，在代数和几何的界面，将被调查。由于提出了跨学科的方法，这一令人兴奋的研究领域的进展有望对理论基础（特别是通过揭示不同领域之间的联系）以及应用（涉及亚椭圆算子的微分方程）产生重大影响。

Many problems and results of harmonic analysis are related to the Laplacian on Euclidean spaces. The Laplacian appears in many important differential equations (describing, e.g., physical phenomena such as heat diffusion, wave propagation or quantum dynamics) and its investigation contributes to the analysis of solutions to these equations (hence to the understanding of said phenomena). A particular focus has been on the relation between boundedness properties of operators in the functional calculus of the Laplacian and smoothness properties of the corresponding spectral multipliers. Despite several exciting breakthroughs in the last decades, many important questions in this area, such as the Bochner-Riesz conjecture, still remain open. Nevertheless basic boundedness properties are fairly well understood, to the extent that robust versions of these boundedness results have been proved, where the Laplacian can be replaced by a more general elliptic operator.Ellipticity, however, is not always a natural assumption. In many contexts, especially in the presence of a sub-Riemannian geometric structure, the natural substitute for the Laplacian need not be elliptic, and it may just be sub-elliptic. Sub-Riemannian geometric structures and sub-elliptic operators are pervasive in many areas of mathematics (e.g., complex analysis and CR geometry, noncommutative Lie groups) and have increasing importance in applications (e.g., in control theory and robotics, and in neurobiology). In this context, even the basic questions about boundedness of functions of a sub-elliptic operator are far from being solved and the known results exploit a mixture of techniques coming from different areas of mathematics (differential geometry, algebra and representation theory, functional and harmonic analysis).The proposed research aims at making substantial progress in the understanding of boundedness properties of functions of sub-elliptic operators and their relations with the underlying geometry, by studying particularly significant examples and by developing more robust techniques. Long-standing open questions of non-Euclidean harmonic analysis, at the interface with algebra and geometry, are to be investigated. Because of the proposed intradisciplinary approach, advances in this exciting research area are expected to have a significant impact on theoretical foundations (especially by shedding light on connections among different fields) as well as in applications (where differential equations involving sub-elliptic operators are used).