Sub-Elliptic Harmonic Analysis

点击翻译按钮获取中文摘要

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).