**Motivation**

Wavelet theory is a highly advanced interdisciplinary field of research, which stretches from pure mathematics to very applied engineering. It has proved to be immensely useful for applied mathematics and engineering like, e.g., image processing (JPEG 2000), EEG and ECG analyses, DNA analysis, climatology, speech recognition, computer vision, etc., but also in pure mathematics. Among its founders the mathematicians Ingrid Daubechies (from Limburg) and Yves Meyer have made significant theoretical contributions relevant to many areas of mathematics, in particular to harmonic analysis and the theory of partial differential equations (PDE theory), on which this research project focuses. The significance of Daubechies’s and Meyer’s results reflects in the numerous prestigious prizes and awards they have received, e.g., the American Mathematical Society Steele Prize (Daubechies), the National Academy of Sciences Award in Mathematics (Daubechies), and the Abel Prize (Meyer).

### FWO Senior Research Grant G022821N: **Noncommutative Wavelet Analysis**

4-year project: 1 Jan 2021 – 31 Dec 2024

### Principal Investigator: Michael Ruzhansky

### Funded Value: **€446,968**

### Postdoctoral Researcher: David Rottensteiner

**Webpage in Dutch**: FWO-onderzoeksproject: Niet-commutatieve wavelet analyse

In this research project we work on advancing the theory of wavelets and its application to mathematical analysis, in particular Calderón-Zygmund operators and pseudo-dfferential operators, in the noncommutative setting, that is, outside the classical theory in the Euclidean space R^n. Our focus of interest specifically connects the works of Daubechies and Meyer to fundamental results in harmonic analysis and PDE theory by Elias Stein (originally from Antwerp), Charles Fefferman (a Fields medalist), and Gerald Folland. Specifically, we plan to make some of their fundamental results in harmonic analysis and PDE theory available in the setting of graded Lie groups. The research project splits into three parts:

**Part I: Wavelet frames and function spaces**

Here we focus on establishing a theory of wavelets on graded Lie groups beyond the setting of stratified Lie groups. The theory of Rockland operators on graded Lie groups is crucial to providing concrete examples of wavelet orthonormal bases in the spirit of Lemarié’s multiresolution approach [Lem89], on the one hand, and the existence of convenient frame generating wavelets as in Führ and Mayeli [FM12], on the other hand. As on R^n, it is of particular importance in applications to provide wavelet frames whose dual frames are systems of wavelet molecules. For non-gradable homogeneous Lie groups, i.e., precisely those homogeneous Lie groups which do not permit a positive Rockland operator, completely new strategies need to be developed. Any advance in this direction will deepen our understanding of harmonic analysis on homogeneous Lie groups substantially.

**Part II:** **Calderón-Zygmund operators and regularity properties**** **

The second part is mainly centered around the works of Meyer, but with strong focuses also on the recent results by Hytönen, on the one hand, and Frazier and Jawerth’s wavelet-based proof of Hörmander’s multiplier theorem, on the other hand. The generalization of the setting of the T(1)-theorem to homogeneous Lie groups has been already established. Some analysis relies on the machinery of Hardy space H1 and its dual space BMO on homogeneous Lie groups, as in the classical monograph Folland and Stein [FS82], Calderón-Zygmund operators on homogenous groups as in the monograph Fischer and Ruzhansky [FR16]. It can be mentioned that the boundedness of Calderón-Zygmund operators on graded Lie groups from 𝐻^1 to 𝐿^1 and from 𝐿^{\infity} to BMO in the special case of pseudo-differential operators was recently established by (non-wavelet methods) in Cardona, Delgado and Ruzhansky [CDR19]: in fact there it follows from the successful generalization of Fefferman’s sharper result [Fef73] on the boundedness of pseudo-differential operators on Hardy spaces and 𝐿^p-spaces.

**Part III: Noncommutative Weyl quantization**

While the analysis of pseudo-differential operators offers enormous potential for the use of wavelets, Gabor frames are the adequate tool to study evolution equations for differential and pseudo-differential operators on R^n (see, e.g., [CTW13,CNR15]). Recent progress on “generalised time-frequency analysis’’ on nilpotent groups, especially orthonormal bases and frames (see [GR18, Ous18, Ous19, GRRV19], link the tools from frame theory to the noncommutative techniques of pseudo-differential operators.

**References:**

[AR20] R. Akylzhanov and M. Ruzhansky. Lp-Lq multipliers on locally compact groups. J. Funct. Anal., 278(3):108324, 49, 2020.

[BPV19] T. Bruno, M. M. Peloso, and M. Vallarino. Besov and Triebel-Lizorkin spaces on Lie groups. Mathematische Annalen, 377, 355-377, 2020.

[CDR19] D. Cardona, J. Delgado, and M. Ruzhansky. Lp-Bounds for Pseudo-Differential Operators on Graded Lie Groups. Preprint, 2019.

[CNR15] E. Cordero, F. Nicola, and L. Rodino. Gabor representations of evolution operators. Trans. Amer. Math. Soc., 367(11):7639-7663, 2015.

[CTW13] E. Cordero, A. Tabacco, and P. Wahlberg. Schrödinger-type propagators, pseudodifferential operators and modulation spaces. J. Lond. Math. Soc. (2), 88(2):375-395, 2013.

[Fef73] C. Fefferman. L^p bounds for pseudo-differential operators. Israel J. Math., 14:413-417, 1973.

[FM12] H. Führ and A. Mayeli. Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization. J. Funct. Spaces Appl., pages Art. ID 523586, 41, 2012.

[FR16] V. Fischer and M. Ruzhansky. Quantization on nilpotent Lie groups, volume 314 of Progress in Mathematics. Birkhäuser/Springer, [Cham], 2016.

[FS82] G. B. Folland and E. M. Stein. Hardy spaces on homogeneous groups, volume 28 of Mathematical Notes. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.

[GR18] K. Gröchenig and D. Rottensteiner. Orthonormal bases in the orbit of square-integrable representations of nilpotent Lie groups. J. Funct. Anal., 275(12):3338-3379, 2018.

[GRRV19] K. Gröochenig, J. L. Romero, D. Rottensteiner, and J. T. v. Velthoven. Balian-Low type theorems on homogeneous groups. Analysis Mathematica, 46, 483-515, 2020

[Lem89] P. G. Lemarié. Base d’ondelettes sur les groupes de Lie stratifiés. Bull. Soc. Math. France, 117(2):211-232, 1989.

[Mar11] A. Martini. Spectral theory for commutative algebras of differential operators on Lie groups. J. Funct. Anal., 260(9):2767-2814, 2011.

[Ous18] V. Oussa. Frames arising from irreducible solvable actions I. J. Funct. Anal., 274(4):1202-1254, 2018.

[Ous19] V. Oussa. Compactly supported bounded frames on Lie groups. J. Funct. Anal., 277(6):1718-1762, 2019.

[tER98] A. F. M. ter Elst and D. W. Robinson. Weighted subcoercive operators on Lie groups. J. Funct. Anal., 157(1):88-163, 1998.