# Noncommutative analysis

Many partial differential equations considered classicaly, particularly boundary problems for domains with simple shapes, exhibit noncommutative groups of symmetries, and noncommutative harmonic analysis arises as a tool in the investigation of these equations. The connection between solving equations on domains bounded by spheres and harmonic analysis on orthogonal groups is one basic case.

The basic method of noncommutative harmonic analysis, generalizing the use of the Fourier transform, is to synthesize operators on a space on which a Lie  group has a unitary representation from operators on irreducible representation spaces. Thus one is led to determine what the irreducible unitary representations of a given Lie group are, and how to decompose a given representation into irreducibles.

From the Introduction of the book:

Decomposition of data (functions, operators) in simple components that can then be efficiently analysed is a powerful approach to a variety of problems. This is the idea behind the Fourier analysis when functions are decomposed into simple waves which leads to a notion of “symbol” – the representation of an operator with respect to these components. In the context of partial differential equations this is known as the theory of pseudo-differential operators which proved to be very effective in the treatment of e.g. elliptic equations with variable coefficients. The important question is how to recapture properties of functions/operators from those of their symbols.

One direction of the research of our group concentrates on the development of a global version of the pseudo-differential type calculus on different types of decompositions such as frame decompositions, spectral expansions, and decompositions given in terms of representations of groups acting on the space. Such phase space analysis is possible in a much wider context than the classical Fourier analysis as our recent research shows:

• in terms of irreducible unitary representations of compact Lie groups

[RT10] Ruzhansky M., Turunen V., Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhauser, Basel, 2010. 724pp. Contents, description and samples, this book and time-frequency analysis, summary at Birkhauser review Bookmetrix

[RT13] Ruzhansky M., Turunen V., Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere, Int. Math. Res. Not. IMRN 2013, no. 11, 2439-2496. arxiv, link

• on nilpotent Lie groups where such a frame is continuous

[FR16] Fischer V., Ruzhansky M., Quantization on nilpotent Lie groups, Progress in Mathematics, Vol. 314, Birkhauser, 2016. xiii+557pp. linkdownload (open access book, first by Imperial College London) Bookmetrix the winner of  Ferran Sunyer I Balaguer Prize 2014

• or in the context of boundary value problems with biorthogonal frames given by a Riesz basis

[RT16] Ruzhansky M., Tokmagambetov N., Nonharmonic analysis of boundary value problems, Int. Math. Res. Notices, (2016) 2016 (12), 3548-3615. offprint (open access), arxiv, link

But, the group structure is actually not necessary! Many of the developed techniques can be extended to settings without group structure, for example:

[DR] Dasgupta A., Ruzhansky M., Eigenfunction expansions of ultradifferentiable functions and ultradistributions, Trans. Amer. Math. Soc., 368 (2016), 8481-8498. arxiv, link

This hugely enlarges the scope of applications of such analysis and concrete physical models imply need to investigate versions of phase space analysis adapted to different underlying geometries of the problems.

“Development of the noncommutative phase space analysis comparable to the Hörmander’s theory of pseudo-differential operators and to the theory of Fourier integral operators, enabling the efficient description of different physical phenomena such as propagation of singularities, transformation of wave front sets, and many others.”

!!!New!!! Rottensteiner D., Ruzhansky M., The harmonic oscillator on the Heisenberg group. C. R. Acad. Sci. Paris, to appear. arxiv

!!!New!!! Kumar V., Ruzhansky M., Hardy-Littlewood inequality and LpLq Fourier multipliers on compact hypergroups. arxiv

!!!New!!! Ruzhansky M., Torebek B., Van der Corput lemmas for Mittag-Leffler functions. II. α-directions. arxiv

!!!New!!! Cardona D., Kumar V., Ruzhansky M., Tokmagambetov N., LpLq boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations. arxiv

Cardona D., Ruzhansky M., Littlewood-Paley theorem, Nikolskii inequality, Besov spaces, Fourier and spectral multipliers on graded Lie groups, arxiv

Fischer V., Ruzhansky M., Fourier multipliers on graded Lie groups, Colloq. Math., to appear. arxiv

Fischer V., Rottensteiner D., Ruzhansky M., Heisenberg-modulation spaces at the crossroads of coorbit theory and decomposition space theory. arxiv

* * *

[KWR20] Kirilov, A.,  de Moraes, Wagner A. A.,   Ruzhansky, M. Partial Fourier series on compact Lie groups, Bull. Sci. Math.  160  (2020), 102853, 27 pp.

[KR20] Kumar, V., Ruzhansky, M. Hausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds. Indag. Math. (N.S.)  31  (2020),  no. 2, 266–276.

[AM20] Akylzhanov, R.,  Ruzhansky, M. Lp-Lq multipliers on locally compact groups. Funct. Anal. 278  (2020),  no. 3, 108324, 49 pp.

[DDR19] Daher, Radouan ;  Delgado, Julio ;  Ruzhansky, Michael . Titchmarsh theorems for Fourier transforms of Hölder-Lipschitz functions on compact homogeneous manifolds. Monatsh. Math.  189  (2019),  no. 1, 23–49.

Many of these and other papers can be dowloaded here.

More posters can be seen here.