Singular integral operators and their boundedness properties are a central theme in harmonic analysis. The group studies these operators in the setting of solvable Lie groups. Here the local part of the operator can be treated with standard methods, and the novelty is the behaviour at infinity. So far, it seems that each such operator poses a new problem and requires its own method. A good deal of this work is carried out together with partners in Sydney and Nancy.
Another setting for singular integrals is Gaussian Harmonic Analysis, where Lebesgue measure in Euclidean space is systematically replaced by a Gaussian measure. The Laplacian is modified accordingly. This can also be considered in infinite dimension. There is no group structure here, but the main difficulty again appears at infinity. The research group has focused on the weak type (1,1) properties of the relevant Riesz operators. This is joint work with mathematicians in Madrid and Genova.
Maximal operators defined in terms of a measure without the doubling property are also considered. Here it is important to find good examples, to understand what happens in general.
Some study is in progress on how boundedness of Fourier multipliers and in general behaviour of the Fourier transform can influence degree of the singularity of the measure.
Persons involved: Ulla Dinger, Rolf Liljendahl, Maria Roginskaya, Peter Sjögren, Anders Öhgren.

Among other problems, we study the functions obtained as integrals against the square root of the Poisson kernel. Here the approach regions are wider than the ordinary nontangential cones.
Persons involved: Fausto Di Biase, Martin Brundin, Jan-Olav Rönning, Peter Sjögren, Olof Svensson.

We are interested in boundary behaviour of solutions to differential equations - which can be described in geometrical terms. E.g. capacities of boundary measures, the Martin boundary (of fractal domains) etc. We are also interested in finding, in a nonsmooth setting, counterparts to classical differential geometry notions such as e.g. curvature, torsion, rotation number, etc. In particular we are interested in finding notions which are meaningful for polygons and similar discrete objects.
Persons involved: Villhelm Adolfsson, Torbjörn Lundh, Maria Roginskaya.

We study representations of semisimple real Lie groups and various generalizations of the related groups and algebras, including quantum groups and Wick algebras.
We are interested and actively conducting research on the following subjects: dynamic quantum groups, special functions related to representations of semisimple and quantum groups; representations of Wick-algebras and constructions of the corresponding C^* q-analogues of the Stone-von Neumann theorem; tensor product and branching of holomorphic representations, Berezin quantization and L^p-harmonic analysis on vector fields on homogeneous space.
Eventually we would like to understand, among other things:

• classification and geometric realization of some interesting classes of representations of those groups and algebras;
• find new algebraic interpretations of special functions and orthogonal polynomials, and use them to study their analytical properties;
• study the L^p-space properties of some interesting group-invariant integral and differential operators; develop isometric representation of Lie groups on Banach spaces.
• applications of operator algebras to quantum groups and to classical harmonic analysis.

Grigori Rozenblioum, together with Michael Melgaard, investigate the spectral properties of the Schrödinger operator with magnetic fields, in particular the estimates and asymptotics of its eigenvalues near the tips of the essential spectrum and convergence and summability of spectral expansions. This is related to more general problems on the relation of spectral properties of the magnetic and nonmagnetic operators in quantum physics. Grigori Rozenblioum also works in global analysis of partial differential and pseudodifferential operators on singular manifolds, first of all index theorems of Atiyah-Singer type. In this direction advanced methods of algebraic topology, K-theory and C^*-algebras are applied.
Johannes Brasche is interested in the problems of classical spectral theory of operators, dealing with description of relations between different self-adjoint extensions of a symmetric operator, where the connection between two sets of boundary conditions for a fixed differential operators is the motivating example.
Persons involved: Johannes Brasche, Michael Melgaard, Grigori Rozenblioum

Problems related to nonlinear hyperbolic partial differential equations are studied. In particular, questions on existence, regularity, well-posedness and asymptotics for solutions to nonlinear wave equations and nonlinear Klein-Gordon equations are treated. Also properties for the scattering operators are focused on.
Persons involved: Philip Brenner, Peter Kumlin

The members of the wavelet group investigate the mathematical and applied aspects of wavelets, a supplement to classical methods of Fourier analysis. The applications pertain to signal analysis, image processing, and numerical approximations of solutions to partial differential equations.