Coherent states and multiparameter singular integrals

A central tool in harmonic analysis is to represent a generic function or operator as a superposition of basic components possessing a certain symmetry. A representation of this kind is often referred to as a representation formula, and allows to study generic functions and operators through properties of their basic components. For example, the Fourier transform provides a systematic way to decompose a function into exponential functions, which in turn allows to obtain representation formulae for basic operators such as the Laplacian. The symmetry of the basic components in a representation formula depends in an essential manner on geometric properties of the function domain or invariance properties of the class of operators. The general aim of this project is to study various fundamental problems on representation formulae for functions and operators in settings that reflect a symmetry with respect to dilations, which are transformations that change the size of an object without changing its shape. For ordinary scalar dilations, such representation formulae have been studied for various classes of function spaces and singular integral operators in wavelet and Calder\`on-Zygmund theory, respectively. However, in numerous applications and settings, more complicated dilation structures appear naturally, for example, multiparameter dilations. Although representation formulae for functions and operators reflecting such a multiparameter dilation structure have individually received considerable attention in recent years, the possible relationship between these topics is still largely unexplored. A particular objective of the project is therefore to develop a systematic theory for obtaining representation formulae for functions that is specific for settings reflecting multiparameter dilation symmetries, and to use the developed theory for the study of multiparameter singular integral operators. The approach towards achieving the project objectives is based on a combination of tools from harmonic analysis and representation theory that are significant for the classes of representations and groups under consideration. The expected results will have a lasting impact on and strengthen the connection between Euclidean harmonic analysis and the representation theory of Lie groups.