Noncommutative Wavelet Analysis

Numerous fields in science and engineering have benefited enormously from the tools of wavelet theory. So far many of these tools are restricted to the (commutative) setting of Euclidean space. In the last couple of years, however, much progress has been made in the area of partial differential equations (PDEs) in noncommuative settings, in particular on compact and nilpotent Lie groups. This project aims at establishing a theory of wavelets on homogeneous, especially graded, nilpotent Lie groups, and to use it for the analysis of hypoelliptic PDEs. The particular importance of graded groups is owed to the fact they permit a remarkable extension of the global differential calculus in Euclidean space, unmatched by many other settings. The project is planned as follows: WP 1 ``Frames and function spaces'' forms the basis of this project by constructing wavelet systems with desirable properties for a wide range of important function spaces on homogeneous, especially graded, Lie groups. WP 2 ``Calderón-Zygmund operators and regularity properties'' employs the wavelet systems from WP 1 to advance the theory of Calderón-Zygmund and pseudo-differential operators on homogeneous and graded groups. WP 3 ``Noncommutative Weyl quantization'' aims at establishing a Weyl quantization of pseudo-differential operators on graded groups, and to study the associated calculus by means of noncommutative short-time Fourier and Wigner transforms, close kin of the noncommutative wavelet transform.