**Seminars in 2006***Tuesday March 14 at 15.15, room MV:L14*

**Genkai Zhang:**Radon, cosine and sine transforms on Grassmannians

**Abstract**: We consider the Radon, cosine and sine transforms on the real, complex and quaternionic Grassmannian manifolds. We find the spectral symbol of the transforms and charaterized their images, by establishing certain Bernstein-Sato type formulas using the Cherednik-Opdam theory. Our results generalize those of Helgason, Alesker-Bernstein and Grinberg. We prove also that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we find the unitary structure of the Stein's complementary series representation realized on the Grassmannians.*Tuesday February 28 at 15.15, room MV:L14*

**Jeff Steif:**Amenability of groups and its connection to the Banach-Tarski Paradox

**Abstract**: The concept of amenability of groups arises in ergodic theory, probability theory and elsewhere. I will motivate and explain this

concept, explain how it relates to a number of different topics and then concentrate on its relationship to the Banach-Tarski paradox. The

possibility of a "Banach-Tarski-type-paradox" in different settings is intimately connected to the nonamenability of certain related isometry

groups. It is for this reason that such paradoxes don't (usually) occur in less than 3 dimensions.*Tuesday February 21 at 15.15, room MV:L14*

**Mikael Persson**: On the Pauli operator with a singular magnetic field

**Abstract**: We study some different self-adjoint Pauli operators corresponding to a singular magnetic field and discuss some physical and mathematical properties one would expect from such operators. We also give some formulas describing the dimension of the kernel of the Pauli operator.*Wednesday February 15 at 15.15, room Mallvinden*

**Henrik Seppänen**: Induced holomorphic representations

**Abstract**: Given a closed subgroup K of a Lie group G and an irreducible representation of K, one can construct a representation of G on a space of sections in a complex vector bundle over the quotient manifold G/K. In the case when G/K is a Hermitian symmetric space one can obtain a representation of G on a space of holomorphic sections. Examples of this construction are the weighted Bergman spaces on the unit disk. We give an outline of this construction, illustrated by the case when G/K is the unit disk. Finally we present a proof by S. Kobayashi of the irreducibility of these so-called induced holomorphic representations.*Tuesday February 7 at 15.15, room MVL:14*

**Peter Sjögren:**Maximal operators for Laguerre functions

**Abstract**: There are several versions of Laguerre functions, each defining an orthogonal basis in L^2 for a suitable measure. This can be in one or several dimensions. For each system there is an associated heat semigroup, with a corresponding maximal operator. We shall examine the boundedness properties of these maximal operators.*Tuesday January 24 at 15.15, room MVL:14*

**Grigori Rozenbioum:**Eigenvalue asymptotics for potential type operators on Lipschitz surfaces

**Abstract**: We establish that integral operators generalizing single layer potential on Lipschitz surfaces obey asymptotic formulas for eigenvalues, natural generalizations of such formulas for smooth surfaces. The reasoning is based on some new approaches in finding eigenvalues estimates and a detailed study of convergence of potential type operators as the Lipschitz surface is approximated by smooth ones.*Tuesday January 10 at 15.15, room MVL:14*

**Andreas Axelsson, Lund**: Quadratic estimates and functional calculi of perturbed Dirac operators

**Abstract**: In this talk I will survey joint work with A. McIntosh and S. Keith, to appear in math. inventiones, on functional calculus of perturbations of the Hodge-Dirac operator d+d* by a bounded, measurable and accretive multiplication operator. I will also discuss some applications and further developments such as the Kato square root problem on Lipschitz domains with mixed boundary conditions.

**Seminars in 2005***Tuesday December 13 at 15.15, room S1*

**Olivia Constantin, Lund**: Hankel operators on Bergman spaces and similarity to contractions

**Abstract**: We investigate the problems of power boundedness, polynomial boundedness and similarity to a contraction for Foguel-Hankel operators on vector-value Bergman spaces.*Tuesday December 6 at 15.15, room S1*

**Yongyang Jin:**Local regularity for solutions of quasilinear elliptic equations under minimal assumptions

**Abstract**: we will discuss some recent results on the regularity of solutions for second-order elliptic differential equations in the linear and non-linear case, especially some minimal requirements for the coefficients and datas.*Tuesday November 29 at 15.15, room S1*

**Nikolay Shirokov, St. Petersburg, Russia**: Lacunary power series with small lacunae and their possible decay rate

**Abstract**: We consider power series of the form $\sum_{k=0}^\infty a_k x^{n_k}$. It is called $(p,A)$- lacunary if $n^k\ge An^p$, $1<p<\infty$. Suppose that the convergence radius of the series equals 1. The question is how fast the sum of the series may decay as $x\to 1$. The history of this question speads over more than 20 years, but only recently sharp estimates were obtained.*Tuesday November 22 at 15.15, room S1*

**Bruno Bongioanni, IMAL-UNL Santa Fe, Argentina**: Sobolev spaces associated to the Harmonic Oscillator

**Abstract**: Hermite Sobolev spaces appears in the literature in connection with Hermite functions expansions. We present a real variable description of these spaces for the whole range of L^p spaces following classical ideas. Two definitions are given: for integer order by means of the natural derivatives in the harmonic oscillator context (annihilation and creation operators), and for any real order using potentials (negative powers of the Hermite operator). As in the classical case, the two definitions coincide for integer order. Some properties are studied using real variable methods. The Hermite Sobolev spaces are natural domains of boundedness for the associated Hermite Riesz transforms and can also be used for studying regularity of solutions of the associated Schrödinger equation.*Tuesday November 8 at 15.15, room S1*

**Joerg Schmeling**, Lund: Limit sets of Kleinian groups and Cauchy type random walks

**Abstract:**In this talk we will illustrate the connection between the dimension of the limit sets of some important Kleinian groups and the transience/recurrence of Cauchy type random walks. These Kleinian groups contain punctured torus groups with badly approximable Thurstons end invariant. We will use the transience of a random walk to give an elementary proof that Kleinian groups may have limit sets of full dimension (a classical result of Bishop and Jones). On the other hand the recently proved Ahlfors conjecture allows to give a purely geometric proof that the standard Cauchy random walk with exponent 2 is stably recurrent.*Tuesday November 1 at 15.15, room S1*

**Grigori Rozenblioum**: Spectral analysis in a mathematical model of the irreversible quantum graph

**Abstract**: A mathematical model is considered, consisting of the partial differential equation with an unusual boundary condition depending on a parameter. The dependence of the spectrum on the parameeter is studied, and a sort of phase transition is found.*Tuesday October 25 at 15.15, room S1*

**Luana Noselli, Milano**: On the maximal operator for the Ornstein-Uhlenbeck semigroup

**Abstract***Tuesday October 18 at 15.15, room S1*

**Lizhong Peng**(Peking University): Compensated compactness and paracommutators

**Abstract**: The compensated compactness is introduced by L. Tartar. Coifman, Lions, Meyer and Semmes give the $H^r-$regularities for many compensated quantities, and show that the compactness (or weak continuity) can be implied from the $H^1-$regularity. In the talk we will show that there is an one to one corresponding between compensated quantities and paracommutators, then the $H^1-$regularity of compensated quantities is equivalent to BMO boundedness of paracommutators, the weak convergence of compensated quantities is equivalent to VMO compactness of paracommutator, the Schatten-von Neumann $S_p-$ property is a natural generalization of VMO compactness, and the theory of paracommutators provides also a ready-made tool for $S_p-$regularity of compensated quantities. It turns out that the cut-off phenomenon of paracommutator exactly coincides with the vanishing moment of the compensated quantity. More natural examples are given by the transvectants.*Tuesday October 11 at 15.15, room S1*

**Mikael Persson:**A particle in a magnetic field of an infinite rectilinear current

**Abstract**: A recent unexpected result by D. Yafaev will be discussed concerning the large-time behavior of a particle in a magnetic field created by an infinite rectilinear current. Both classical and quantum mechanics situations will be presented.*Tuesday September 20 at 15.15, room S1*

**Grigori Rozenblioum:**Spectral properties of the Pauli operator with weekly perturbed constant magnetic field

**Abstract**: The Pauli operator with constant magnetic field has spectrum consisting of eigenvalues with infinite multiplicity, called

Landau levels. We describe what happens with these eigenvalues when the field undergoes a week perturbaltion. among other things, a relation with Toeplitz operators in the Segal-Bargmann-Fock space will be explained.*Tuesday September 13 at 15.15, room S1*

**Genkai Zhang**: Radon transform on symmetric spaces.

**Abstract**: We consider Radon transform on the Grassmannian manifolds of r-dimensional $K$-subspaces in $K^n$, where $K$ is the field of real, complex or quanternionic numbers. We find an inversion formula and answer some question of Grinberg and Rubin. Some related questions for non-compact symmetric spaces will also be discussed.*Tuesday May 24 at 15.15, room S1*

**Evgeni Doubtsov**, S:t Petersburg: The uncertainty principle, the Boole formula, and Hankel operators

**Abstract***Tuesday 26 April at 15.15, room S1*

**Adam Nowak**, Wroclaw: Weighted estimates for the Hankel transform transplantation operator

**Abstract**: The Hankel transform transplantation operator is investigated by means of a suitably established local version of the Caldero'n-Zygmund operator theory. This approach produces weighted norm inequalities with weights more general than previously considered power weights. Moreover, it also allows to obtain weighted weak type (1,1) inequalities, which seem to be new even in the unweighted setting. As a typical application of the transplantation, multiplier results in weighted L^p spaces with general weights are obtained for the Hankel transform of any order greater than -1 by transplanting cosine transform multiplier results.*Tuesday 19 April at 15.15, room S1*

**Katerina Nemcova**: Approximation by point potentials in the presence of a magnetic field

**Abstract***Tuesday 12 April at 15.15, room S1*

**Peter Sjögren**: Generaliserade Poissonintegraler och svaga L1-uppskattningar

**Abstract**: Poissonintegralen och flera varianter av den kan uppskattas och ofta karakteriseras i termer av svaga L1. Vi skall ge en översikt av några resultat av detta slag. De flesta är från 1970- och 80-talen men har nyligen kommit till användning i samband med bl a Laguerrefunktioner.*Tuesday 29 March at 15.15, room S1*

**Johannes Brasche**, Clausthal, Tyskland: Inverse spectral theory for symmetric operators with several gaps

**Abstract**: Let S be a symmetric operator in a Hilbert space. Suppose that there exists a non-empty open set of real numbers which is contained in the resolvent set of at least one self-adjoint extension of S. We shall discuss the question about what kinds of spectra the other self-adjoint extensions of S (if there are any) can have within the mentioned open set of real numbers. The results presented in the talk will be based on joint work with S.Albeverio, M.M.Malamud and H.Neidhardt. A main tool in our investigations will be the Weyl functions of the self-adjoint extensions. These functions belong to the Nevanlinna class and via the Nevanlinna theory one gets an integral representation for the Weyl functions. The measure in the integral representation of the Weyl function of a self-adjoint extension A of S stores the complete information about the spectrum of A.*Tuesday 15 March at 15.15, room S1***Bent Orsted**(Aarhus, Denmark): Logarithmic Sobolev inequalities on the Heisenberg group**Abstract**: For the intrinsic geometry on the Heisenberg group and its natural compactification we obtain an analogue of the logarithmic Sobolev inequality; such inequalities have a wide range of applications in differential geometry, analysis, and physics.**Khalid Koufany**(Nancy, France): The rotation number for the conformal groups**Abstract**: Let $ G/K$ be an irreducible Hermitian symmetric space of tube type, and $S$ its Shilov boundary. We use the Maslov cocycle defined on $S$ to construct an invariant on the conformal group $G$ which generalize the symplectic rotation number.*Tuesday 8 March at 15.15, room S1***Grigori Rozenblioum**: Zero modes of the Pauli operator and related problems in Function Theory**Abstract**: We describe conditions on the function F(z) ensuring that the space of entire functions f(z) quadratically

integrable with weight exp(F(z)) is infinite-dimensional. The conditions are expressed in the terms of the Laplacian of F(z).

The results are applied to the study of the spectrum of the Pauli operator describing the electron in a magnetic field*Tuesday 22 February at 15.15, room S1*

**Genkai Zhang**: Radon inversions on Grassmannians

**Abstract**: I'll report the paper by Grinberg and Rubin (Ann. Math. 2004) where they found an inversion formula for the Radon transform of functions on Grassmanian of $k$-dimensional subspaces in $R^n$ to functions on $l$-dimensional subspaces, which is defined by integrating over the set of $k$-subspaces contained in a given $l$-subspaces.*Tuesday 1 February at 15.15, room S1*

**Lyudmila Turowska**: On the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups

**Abstract**: Arveson (1974) discovered a connection between the invariant subspace theory and spectral synthesis. He defined (operator) synthesis for subspace lattices and proved the failure of operator synthesis by using the famous example of Schwartz on non-synthesizability of the two-sphere $S^2$ for $A({\mathbb R}^3)$. Froelich made this connection more precise for separable abelian group. We generalise this result to second countable locally compact groups $G$. Namely, we prove that a closed subset $E\subset G$ is set of local spectral synthesis for $A(G)$ iff the diagonal set $E^*=\{(s,t)\in G\times G\mid st^{-1}\in E\}$ is a set of operator synthesis with respect to Haar measure. We give simple proofs that one-point set is spectral and any closed subgroup of second countable group is a set of local spectral synthesis. We will also discuss a connection between Ditkin sets and operator Ditkin sets. We shall start with discussion on the Fourier algebras for general locally compact groups (due to Eymard) and the notion of spectral synthesis for them. Concrete important examples of sets of synthesis will be given. Then we proceed with the notion of operator synthesis, its historical background and its application to harmonic analysis.*Tuesday 11 January at 15.15, room S1*

**Maria Roginskaya**: About L-p improving properties of some singular measures

**Abstract**: There is a general rule, that a convolution of functions has "improved" behaviour in comparison with the functions themselves. The same is true for a convolution of function with some measures, even singular measures. I'm going to present an explanation on how this can be.

**Seminars in 2004***Tuesday 21 December at 15.15, room S1*

**Leo Larsson**, Uppsala: Inequalities of Carlson Type with Applications

**Abstract***Tuesday 7 December at 15.15, room S1*

**Salem Ben Said**(Aarhus, Denmark): On Bessel functions and Dunkl operators - Theory and applications*Tuesday 30 November at 15.15, room S1*

**Marcus Sundhäll**: Schatten-von Neumann properties of bilinear Hankel forms of higher weights and some related matrix-valued Bergman projections.

**Abstract**: We study the Schatten-von Neumann properties of Hankel forms of higher weights on the unit ball of $C^n$. It turns out that some questions are closely related to boundedness of matrix-valued Bergman type projections. I shall present some preliminary results for the Bergman projections on Bergman spaces of tensor-valued holomorphic functions on the unit ball of $\mathbb{C}^n$. More precisely I will characterize bounded, compact, Hilbert-Schmidt and Schatten-von Neumann class $\mathcal{S}_p$-Hankel forms in terms of the membership of the symbols in certain Besov spaces, $2<p<\infty$, using boundedness of some related projections.*Tuesday 9 November at 15.15, room S1*

**Andreas Juhl**, Uppsala: Families of conformally invariant differential operators

**Abstract***Tuesday 26 October at 15.15, room S1***Sorina Barza**, Karlstad: Two-dimensional decreasing rearrangements and the corresponding Lorentz spaces*Tuesday 19 October at 15.15, room S1*

**Khalid Koufany**, Nancy, Frankrike: A cohomology class associated with the Maslov cocycle on Hermitian symmetric spaces*Tuesday 12 October at 15.15, room S1*

**Johannes Brasche**: Interactions along Brownian paths*Tuesday 5 October at 15.15, room S1***Wolter Groenevelt**: Racah coefficients and Wilson functions*Tuesday 21 September at 15.15, room S1*

**Grigori Rozenblioum**: Zero modes for the Pauli operator with singular magnetic fields; application of entire functions**Abstract.**The Pauli operator describes the behavior of quantum particles with spin 1/2 in the presence of a magnetic field, in non-relativistic regime. It is well known that in dimension 2, for a compactly supported sufficiently regular magnetic field, the Pauli operator has eigenfunctions with eigenvalue zero, 'zero modes', and their number is determined by the total flux of the field (Aharonov-Casher theorem). We consider fields with an infinite flux, generated by a system of very singular, Aharonov-Bohm type, magnetic vortices placed at the points of some infinite discrete set E in the plane. Using the theory of entire functions we establish that, under certain conditions, if the set E is not too large, the Pauli operator has an infinite-dimensional zero subspace. On the other hand, if this set is not too small, the point zerto of the spectrum is well separated from the rest of the spectrum. These results might be useful for quantum computing.*Tuesday 14 September at 15.15, room S1*

**Véronique Fischer**: Study on Some Two-Step Nilpotent Lie Groups

**Abstract**

Tuesday 7 September at 15.15, room S*1*

**Peter Sjögren**: Functional calculus for the Ornstein-Uhlenbeck operator

**Abstract.**The Ornstein-Uhlenbeck operator L is a self-adjoint Laplacian connected with the Gaussian measure in Euclidean space. Its spectrum is the set of natural numbers. Let m be a function defined on this spectrum. Then m(L) is bounded on L^p for the Gaussian measure if m has a holomorphic extension to a cone |arg z| < b, with b large enough. The sharp minimal value of b is known, and for this value of b one also needs Mihlin-type conditions on the boundary. We shall see that one can weaken the conditions by translating the cone to the right. The proof goes via estimates for the imaginary powers of L. This is joint work with Mauceri and Meda, and the earlier parts also involve Garcia-Cuerva and Torrea.

*Tuesday 31 August at 15.15, room S1*

**Toshio Horiuchi**, Ibaraki University, Japan: Missing Terms in Hardy-Sobolev Inequalities and its Application

**Abstract***Tuesday 24 August at 15.15, room S1*

**Shayne Waldron**, Auckland: An introduction to tight frames

**Abstract.**Frame representations are useful because they are technically similar to orthogonal expansions (they simply have more terms) and can be constructed to have desirable properties that may be impossible for an orthogonal basis, e.g., in the case of wavelets certain smoothness and small support properties. I will give an elementary introduction to tight frames and talk about some of my recent work. This deals with tight frames which share some of the symmetries of the underlying space (which an orthogonal basis cannot express). One important example is orthogonal polynomials of several variables for a weight which has some symmetries.*Tuesday 17 August at 15.15, room S1*

**Keith Rogers**: Sharp van der Corput estimates and minimal divided differences

**Abstract.**I will find the sharp constant in a sublevel set estimate which arises in connection with van der Corput's lemma. I will also find the sharp constant in the first instance of the van der Corput lemma. With these bounds I will improve the constant in the general van der Corput lemma, so that it is asymptotically sharp.*Wednesday 9 June at 15.15, room S2*

**Takaaki Nomura**, Kyoto, Japan: A characterization of symmetric tube domains by convexity of Cayley transform images.

*Tuesday 8 June at 13.00-15.00, room S1*

**Hitoshi Ishii**, Waseda University, Tokyo, Japan: Convexified Gauss curvature flow and the wearing process of a stone

**Abstract.**Professor Ishii is well known for his decisive input to the modern theory for non-linear PDE in non-divergence form. The primary virtues of this theory are that it allows merely continuous functions to be solutions of fully nonlinear equations of second order, that it provides very general existence and uniqueness theorems and that it yields precise formulations of general boundary conditions. The present talk is devoted to a particular non-linear problem related to the movement of a surface with local normal velocity proportional to its Gauss curvature.

Wednesday 2 June at 16.30, room S1

**Kaj Nyström**, Department of Mathematics, Umeå University: Square Functions, Uniform Rectifiability and Regularity of Parabolic Free Boundary Problems

**Abstract**

*Tuesday 18 May at 15.15, room S1*

**Daniel Levin**, Technion, Haifa: On the spectrum of the Dirichlet Laplacian on broken strips

**Abstract**

*Tuesday 4 May at 15.15, room S1*

**V.Molchanov**: Canonical representations and overgroups for hyperboloids**Abstract**

*Tuesday 20 April at 15.15, room S1*

**Adam Nowak**, Wroclaw: On Riesz transforms for Laguerre expansions

**Abstract.**We prove that Riesz transforms and conjugate Poisson integrals associated with the multi-dimensional Laguerre semigroup are bounded in L^p, 1<p<\infty . Our main tools are appropriately defined square functions and the Littlewood-Paley-Stein theory.

Tuesday 30 March at 15.15, room S1

**Ekaterina Shulman**: On some functional equations and representations of topological.

**Abstract.**

Tuesday 23 March at 15.15, room S1

**Peter Sjögren**: Kärnor för Rieszoperatorer i en lösbar Liegrupp av 3x3-matriser.

**Abstract.**Matrisgruppen SL(3,C) av komplexa 3x3-matriser med determinant 1 har en lösbar delgrupp NA av triangulära matriser. Vi börjar med en beskrivning av dessa och några relaterade matrisgrupper, och skall sen komma fram till kärnor för Rieszoperatorer på NA. Dessa kärnors beteeende i oändligheten har hittills inte studerats i något fall som detta, där rangen är två.

*Tuesday 16 March at 15.15, room S1***Henrik Petersson**: On the Hypercyclicity Criterion. Fristående fortsättning från den 2 mars.

**Abstract:**Abstract. In a previous talk we discussed Godefroy-Shapiro's Theorem (91'): Every convolution operator, not a scalar multiple of the identity, is hypercyclic on the Frechet space of entire functions in n-variables. Recall, a continuous linear operator T on a TVS X is hypercyclic (or more suggestive, universal) if there is a (hypercyclic/universal) vector f such that the orbit Orb(T,f)={f,Tf,T^2 f,...} is dense. We saw that there is a simple proof of G-S Theorem resting on the famous Hypercyclicity Criterion:(HC) Let T be a continuous linear operator on a separable Frechet space X. Assume there are dense subsets Z,Y and a map S:Y \to Y such that

(i) T^n \to 0 poinwise on Z,

(ii) S^n \to 0 pointwise on Y,

(iii) TS = identity on Y.

Then T is hypercyclic.In the talk we shall prove the HC by proving a more general statement, also known as the hypercyclicity criterion. We motivate this by the fact that it has been conjectured whether the HC is in fact necessary. Now, it has been shown that the "weak" HC stated above is not necessary but it is an open problem if the more general version is necessary.

*Tuesday 2 March at 15.15, room S1*

**Henrik Petersson**: Hypercyclic Operators

**Abstract**: A continuous linear operator T:X \to X is hypercyclic if there is a (hypercyclic) vector f\in X such that the orbit Orb(T,f)={f,Tf,T^2f,...} is dense. A famous theorem, due to Godefroy & Shapiro (1991), states that every non-constant convolution operator, on the space H of entire functions in n-variables, is hypercyclic. On the other hand, there are few examples of hypercyclic non-convolution operators. However, recently we were able to establish some classes of such operators by applying results from our study of PDE-preserving operators, i.e., operators that map kernel-sets of convolution operators invariantly. In the talk we shall discuss the (beautiful) proof of Godefroy-Shapiro's Theorem, and the ideas of how we can obtain hypercyclic non-convolution operators on H. However, we start with the definitions and thus do not assume the listeners are familiar with the notion of hypercyclicity before.

*Tuesday 24 February at 15.15, room S1*

**Johannes Sjöstrand**, Palaiseau: The Calderón problem with partial data (joint work with C. Kenig and G. Uhlmann).

**Abstract.***Tuesday 17 February at 15.15, room S1*

**Lyudmila Turowska**: Operator synthesis, spectral synthesis amd linear operator equations. (continuation of my talk on Tuesday 3/2)*Tuesday 3 February at 15.15, room S1*

**Lyudmila Turowska**: Operator synthesis, spectral synthesis amd linear operator equations.

**Abstract**: W.Arveson introduced the notion of synthesis for operator algebras and subspace lattices in 1974 in connection with some problems of the Invariant Subspace Theory establishing an important relation with spectral synthesis for locally compact abelian group. We extend this interplay to include other topics such as harmonic analysis for tensor algebras approximation theory, linear operator equations and spectral theory of multiplication operators in the space of bounded operators and symmetrically normed ideals of operators. The talk is based on a joint work with Victor Shulman.*Tuesday 27 January at 15.15, room S1***Viktor Kolyada**, Karlstad: Estimates of rearrangements and embedding theorems

**Abstract**: We consider non-increasing rearrangements of functions of several variables. We study estimates of the rearrangement of a given function in terms of its derivatives and moduli of continuity. We apply these estimates to prove some Hardy-Littlewood and Sobolev type inequalities*Tuesday 20 January at 15.15, room S1*

**Ljudmila A. Bordag**, Halmstad: Quasi periodic vortex structures in two-dimensional flows in an inviscid incompressible fluid.

**Abstract**

**Seminars in 2003***Tuesday 16 December at 15.15, room S1***Katerina Nemcova**, Nuclear Physics Institute, Prag: Approximation by point-interaction Hamiltonians in dimension two

**Abstract.**

*Tuesday 9 December at 15.15, room S1*

**Grigori Rozenblioum**: On spectral properties of a perturbed multi-vortex Aharonov-Bohm Hamiltonian.

**Abstract***.*Aharonov-Bohm Hamiltonian is the Schrödinger operator with a very singular magnetic field. 45 years ago introduction of such Hamiltonians lead to some quantum paradoxes, which contributed to development of gauge theories. We consider the Aharonov-Bohm operator with a finite or infinite system of singular magnetic fluxes and establish the diamagnetic inequality and proper versions of the Hardy inequality. Those are used to find estimates and asymptotics of the discrete spectrum of this operator perturbed by an electric potential.*Tuesday 2 December at 15.15, room S1*

**Andrei Shkalikov**, Moscow State University: A short proof of the Pontrjagin-Krein-Langer theorem on invariant subspaces.

**Abstract**

*Tuesday 2 December at 16.15, room S1*

**Mark Malamud**, Donetsk: Inverse problems for Hamiltonian systems and matrix Sturm-Liouville equation.

**Abstract***Tuesday 25 November at 15.15, room S1***Jeff Steif**: Influence of variables. Part II (This will be a continuation of last week's talk where the main theorem will be proved)*Tuesday 18 November at 15.15, room S1***Jeff Steif**: Influence of variables. Part I (Joint analysis and statistics seminar)*Tuesday 11 November at 15.15, room S1*

**Christer Borell**: Sharp geometric bounds on the measures

**Abstract**

*Tuesday 4 November at 15.15, room S1*

**Håkan Blomqvist**: Om rum-tidmedelvärden för lösningar till ickelinjära Klein-Gordonekvationer

**Abstract***Tuesday 21 October at 15.15, room S1*

**Vladimir A. Mikhailets**, Kiev: Common Eigenvalue problem and periodic Schrödinger operator

**Abstract***Tuesday 14 October at 15.15, room S1*

**Genkai Zhang**: Spherical transform of the canonical kernels on root systems of type BC.

**Abstract**: We consider a root system of type BC with general (not necessarily integral) multiplicity. We compute the spherical transform of the canonical kernels using the represenations of the Hecke algebras studied by Opdam. This generalizes earlier results of Upmeier-Untergerber and myself.

*Monday 6 October at 15.15, room S2***Patrick Ostellari**: A new proof of the heat kernel estimate on the real hyperbolic disc.

**Abstract**: In a recent joint work with J.-Ph. Anker, we proved, by means of elementary methods (up to technical "details"), a global estimate for the heat kernel on any Riemannian noncompact symmetric space. During the talk, we shall present this method in the simplest nontrivial case: the one of the 2-dimensional real hyperbolic space, which yields another proof of Davies & Mandouvalos' classical result.

*Tuesday 30 September at 15.15, room S1***Maria Roginskaya**: Singularity of vector valued measures (joint with M. Wojciechowski)

**Abstract**: In this talk I will show that even a relatively weak restriction on the direction of Fourier transform of a vector valued measure can affect the Hausdorff dimension of the measure (i.e. its level of singularity). This is a new sort of "Uncertainty Principle"-type result, which doesn't occure for the scalar valued case.*Tuesday 23 September at 15.15, room S1*

Maria Roginskaya

**Abstract**: I will present a series of results, which extends the formula for the energy of a positive finite measure in Euclideanan space via its Fourier transform to the case of a signed measure on a torus and even any compact Riemannian manifold. This formula allows to get a quick progress in such classical questions about singular measures as Hausdorff dimension and L^p-improving properties.*Monday 8 September at 15.15, room S1*

**Silvia Verzeletti**, Milano: An introduction to square functions.

**Abstract**

*Tuesday 9 September at 15.15, room S1*, Essen: Existence of the best constant for the Strichartz inequality.

Markus Kunze

**Abstract***Tuesday 26 August at 15.15, room S1*: On the Lewis functional equation

Joachim Hilgert

**Abstract**

*Tuesday 2 September at 15.30, room S1*

**Toshio Horiuchi**, Ibaraki University, Japan: Two topics on Quasilinear degenerate elliptic equations; Removable singularities of solutions, Blow-up of Minimal solution.

**Abstract**: a bounded smooth domain of $\Bbb R^N$. We shall explain some results on quasilinear degenerate elliptic equation. We shall treat the operator given by A_p(u)= -\mathrm{div}(A(x)|\nabla u|^{p-2}\nabla u). Here $p\in (1,+\infty)$, $A(x)\ge 0$. If $A(x)\equiv 1$, this is called $p$-harmonic noperator. Hence we put L_p(u)= -\mathrm{div}(|\nabla u|^{p-2}\nabla u). In the first part, we treat the equations with nonlinear terms in the left hand side as absorption term. We study removable singularities of solutions and the unique existence of bounded solutions for genuinely degenerate elliptic equation. In the second part we treat the equations with a positive nonlinearity in the right hand side. In connection with combustion theory and other applications, we are interested in the study of positive minimal solutions.*Wednesday 11 June at 15.15, room S2*

**Yurif M. Berezansky**(Kyiv): Generalized selfadjoint operators

**Abstract**. We introduce operators that act from a positive space into a negative space into some Hilbert rigging and that are Hermitean in the sense of the zero space or that are selfadjoint in some natural generalized sense. We investigate perturbations of such operators and generalized eigenvector expansions. Now it is possible to consider Schrödinger operators with a potential that is a generalized function with an arbitrary support.*Tuesday 3 June at 15.15, room S1*

**Lennart Frennemo**: Saturationsproblem för en klass av faltningar

**Abstract**. Det är välbekant att t ex Laplacetransformen av en funktion inte kan avta för snabbt utan att funktionen är identiskt noll, åtminstone på ett intervall. Denna typ av problem visas allmänt för en klass av faltningar som bl a innehåller Laplace-, Meijer- och Weierstrasstransformerna. Resultaten kan även generaliseras till n-dimensioner.

*Friday 6 June at 13.15, room S1*, Shimane University, Japan: Fatou and Littlewood theorems for Poisson integrals with respect to non-integrable kernels

Hiroaki Aikawa

*Tuesday 20 May at 15.15, room S1*

**Johannes Friedemann Brasche**: Interactions along Brownian paths

**Abstract**. We shall discuss the spectra of quantum mechanical Hamiltonians with potentials absolutely continuous with respect to the occupation time measure of a Brownian time motion.*Tuesday 13 May at 15.15, room S1*

**Peter Sjögren**: Vektorvärda Fouriermultiplikatorer och faltningsoperatorer

**Abstract**. Några resultat ur T. Hytönens avhandling och deras bakgrund skall presenteras. Fouriermultiplikatorer och singulära integralkärnor är definierade i det Euklidiska rummet, men tar värden som är operatorer mellan två Banachrum. Det gäller att finna geometriska villkor på Banachrummen och villkor på operatorerna som gör det möjligt att utvidga kända resultat från det skalärvärda fallet.*Tuesday 6 May at 15.15, room S1*

**Katerina Nemcovà**, Prag: Magnetic layers with periodic point perturbations

**Abstract**. We study spectral properties of a spinless particle confined to a Dirichlet layer and interacting with periodic point potentials and a homogeneous magnetic field perpendicular to the layer. Provided that the magnetic flux through the elementary cell is rational, Landau-Zak transformation and Krein's formula yield a description of the spectral bands.*Tuesday 29 April at 15.15, room S1*

**Peter Sjögren**: Fouriermultiplikatorer och singulära integraler enligt Hytönen

**Abstract:**T. Hytönen har just disputerat i Helsingfors på en uppseendeväckande bra avhandling om främst vektorvärda Fouriermultiplikatorer och faltningsoperatorer. I detta föredrag skall jag presentera hans vackra skärpning av såväl Hörmanders som Mihlins välkända multiplikatorsatser, i det klassiska, skalärvärda fallet.

*Tuesday 22 April at 15.15, room S1*

**Per Hörfelt**: Analytical Tools in Option Pricing

**Abstract**. This talk gives a brief introduction to the theory of option pricing and discusses how some result from mathematical analysis can be useful in option pricing. To be more specific, the talk will show how the Rosenthal inequality, the isoperimetric inequality for Wiener measure, and the Krein condition can be applied in the pricing of certain path-dependent options.*Tuesday 1 April at 15.15, room S1*

**Grigori Rozenblioum**: Spectrum of boundary value problems for Dirac operator with Coulomb potential

**Abstract**. The coulomb interaction for the Dirac operator is, probably, the most simple example of a very singular perturbation, coming from nature. This perturbation is not relatively compact, and therefore even the definition of the operator itself encounter certain obstackles. We study some boundary value problems for this operator and describe spectral properties.

*Tuesday 25 March at 15.15, room S1*

**Vladimir A. Mikhailets**, Kiev: Spectral properties of a self-adjoint elliptic operator over a bounded domain

**Abstract**: The eigenvalue distribution for a self-adjoint elliptic operator with general (possibly non-local) boundary conditions on a smooth bounded domain is studied.*Tuesday 4 March at 15.15, room S1*, Lille: Moment Problems on Semi-Algebraic Sets and Applications

Florian Vasilescu

**Abstract**: The K-moment problem for a semi-algebraic compact set K in the real and complex n-dimensional Euclidean spaces is discussed. Our method allows a significant reduction of the necessary positivity conditions, as well as a good control of the support of the representing measures. The counterparts of these results for Hilbert space operator data are also presented. Applications to the trigonometric moment problem with operator data are given and some connections with the existence of joint unitary dilations for commuting multioperators are emphasized.*Tuesday 25 February at 15.15, room S1***V.Nazaikinskii**: On the propagation of electromagnetic waves in ionosphere

**Abstract**: We study the propagation of electromagnetic waves in an ionosphere layer. Results are obtained both for the light region (the geometrical optics

approximation) and the shadow region. For specific ionosphere models occurring in practice, exact formulas are obtained including reflection, penetration and

channeling phemomena.*Tuesday 21 January at 15.15, room S1*

**G. Rozenblioum**: Riesz L_p summability of spectral expansions for the magnetic Schrödinger operator

**Abstract**: For any selfadjoint operator in the space L_2, spectral expansion of a function f in L_2 converges to f in _2. However, if f belongs to L_p, for p being not equal to 2, such expansion does not converge in L_p, with very few exceptions. Convergence usually is improved id one considers Riesz means of the spectral expansion. We consider spectral expansions corresponding to a Schrödinger operator with constant magnetic field and prove that Riesz means of proper order

converge in L_p.This problem is related to analysis on non-isotropic Heisenberg groups.*Tuesday 14 January at 16.00, room S1***Andreas Axelsson**, Australian National University, Canberra: Transmission problems for Dirac's and Maxwell's equations with Lipschitz type interfaces. Abstract

**Seminars in 2002***Tuesday, November 19 at 15.15 room S1*

**Kaj Nyström**, Föreningssparbanken: Big pieces of graphs and Caloric measure in parabolic flat domains

**Abstract**: In my talk I will describe joint with Steve Hofmann and John Lewis on the caloric measure on what we refer to as parabolic chord arc domains. In particular I will show that in a parabolic chord arc domain with vanishing constant, the logarithm of the density of caloric measure with respect to a certain projective measure is of vanishing mean oscillation. A partial converse of this result will also be discussed.*Tuesday, November 12 at 15.15 room S1*

**Genkai Zhang**: Biorthogonal expansion of Cayley transform of Jack symmetric functions.

**Abstract**: Macdonald and Koornwinder introduced a family of remarkable orthogonal polynomials associated to any root system (of a simple Lie algebra) with general multiplicity. For root system of type A those polynomials are the Jack symmetric polynomials (generalizing the Schur character formula and spherical polynomials on symmetric cones) and Meixner type polynomials. We find an biorthogonal expanison of the Cayley transform of the symmetric (and non-symmetric) Jack functions in terms of those polynomials, and we study their applications. (Joint work with S. Sahi).

*Tuesday, November 5 at 15.15 room S1*

**M. Englis**, Prag: Operator models on bounded symmetric domains

**Abstract***Tuesday, October 22 at 15.15 room S1*

**Fausto Di Biase**, Universitá 'G.d'Annunzio' (Pescara, Italien): A potential theoretic approach to twisting

**Abstract:**

We give a new, potential theoretic approach to the study of twist points in the boundary of plane domains.*Thursday, October 17 at 15.15 Hörsalen*

**Aline Bonami**, Orleans, som besöker oss för att promoveras till hedersdoktor vid GU:

bones and directional asymptotic self-similarity

*Tuesday, October 15 at 15.15 room S1*

**Giancarlo Mauceri**, Genova: Holomorphy of spectral multipliers of*L*^{p}

**Abstract**

*Tuesday, October 1 at 15.15 room S1*

**Georgi Popov**, Nantes: Quantum resonances for transparent obstacles

**Abstract**

*Tuesday, September 24 at 15:15 room S1*

**Andrei Khrennikov**, Växjö: P-adic Partial Differential Equations and their Applications to Physics

**Abstract:**

We shall discuss the role of number field (e.g. real, complex, p-adic or algebraic extensions of p-adic numbers) in physics, in particular, comology and string theory. Then there will be discussed mathematical problems induced by p-adic theoretical physics: distributions on p-adic spaces with p-adic and complex vales, Fourier (and Laplace) transform, Cauchy problem, pseudo-differential operators. References: 1. Khrennikov A.Yu., p-adic valued distributions and their applications to the mathematical physics, Kluwer Acad. Publishers, Dordreht/Boston/London, 1994. 2.Khrennikov A.Yu., Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models. Kluwer Acad. Publishers, Dordreht/Boston/London, 1997.*Tuesday, September 3 at 15:15 room S1*

**Analysseminariet****Daniel Levin**: On isoperimetric dimensions of product spaces**Abstract**. It is well-known that dimensions of Euclidean spaces add up, if one considers their product,**R**^{d}=**R**^{m}x**R**^{n}, d=m+n. For Riemannian manifolds, the notion of dimension is more delicate, e.g. the topological dimension does not reflect their geometry at infinity. However, one may introduce*an isoperimetric dimension*through isoperimetric inequalities. The dimension introduced in this way is not a number but a family of functions indexed by a parameter p, 1<p<\infty. Our main result generalizes the addition of dimensions in the euclidean case using the notion of the isoperimetric dimension.

*Monday, August 26 at 10:00 room S1*

**Analysseminariet****Michael Demuth**TU Clausthal - Zellerfeld, Germany: On the equilibrium potentials in mathematical scattering theory**Abstract**. In scattering theory by hard cores the perturbed operator is determined by Dirichlet conditions on the boundary of the obstacle. Due to Dynkin' s formula the corresponding resolvent difference can be represented stochastically by the equilibrium potential. The scattering system is complete if the equilibrium potential is integrable, or if the obstacle has finite capacity. In case of two obstacles the difference of the scattering matrices can be estimated by the capacity of the symmetric difference of the obstacles. This can be used to study the more realistic situation of high potential barriers.**Michail Solomyak**Weizmann Institute, Rehovot, Israel: On the spectrum of the Laplacian on metric graphs**Abstract**. A metric graph is a graph whose edges are viewedas line segments of positive length, rather than just pairs of vertices. The Laplacian on such graph is the operator of second derivative on each edge, complemented by the Kirchhoff matching conditions at vertices. The spectrum of the Laplacian can be quite different, reflecting geometry of a given graph. Recent results on this subject will be reported. The most detailed results concern a special case of graphs, namely the so-called regular trees.*Wednesday, June 12 at 13:15 room S1*

**Analysseminariet****Yu.M.Berezansky**, Kyiv, Ukraine: Some generalizations of the classical moment problem.*Tuesday, June 11 at 15:15 room S1*

**Analysseminariet****Michel Zinsmeister**, Orleans: Hausdorff dimension of Julia sets as a function of the polynomial.**Abstract**. The aim of this talk is to survey the properties of the function assigning to a polynomial P the Hausdorff dimension of its Julia set. We will briefly describe the thermodynamic formalism leading to the understanding of the hyperbolic case. We will then describe the parabolic implosion phenomenon which is the key to the study of discontinuities at bifurcation points. We will show in particular a new proof of Shishikura's theorem asserting that HD is generically equal to 2 for the parameters belonging to the boundary of the Mandelbrot set.*Thursday, June 6 at 10:15 room S1*

**Analysseminariet****Michael Gnewuch**: Differentiable L^{1}-functional calculus for self-adjoint operators*Tuesday, June 4 at 15:15 room S1*

**Analysseminariet****Pekka Koskela**, Jyväskylä: Mappings of finite distortion**Abstract**. Mappings of finite distortion can be considered as a generalization of analytic functions. If the distortion function is suitably integrable, then these mappings have many nice properties such as discreteness and openness. I will review the recent work on this area.*Tuesday, May 21 at 15:15 room S1*

**Analysseminariet****Michael Melgaard**: On bound states for a system of weakly coupled Schrödinger equations in one space dimension (work in progress).*Tuesday, May 7 at 15:15 room S1*

**Analysseminariet****Jaak Peetre**, Lund: Comments on Lions's formula for the reproducing kernel of some spaces of harmonic functions*Tuesday, Maj 14 at 15:15 room S1*

**Analysseminariet****Johannes Sjöstrand**, Palaiseau: Bohr-Sommerfeld quantization-conditions in dimension 2.**Abstract**: The Bohr-Sommerfeld quantization condition often permits to find all eigen-values in certain regions, for self-adjoint differential operators in dimension 1 (in various asymptotic limits), but the general wisdom is that in higher dimensions we encounter severe limitations. It is quite remarkable then that for fairly wide and stable classes of non-selfadjoint operators in dimension 2 we get results that are analogous to the classical ones in dimension 1 in the self-adjoint case. We describe these results starting with a joint work with A.Melin, and similiar results for perturbations of non-selfadjoint operators, partly in collaboration with M. Hitrik.*Tuesday, April 23 at 15:15 room S1*

**Analysseminariet****Johannes Sjöstrand**, Palaiseau: Remarks about spectrum and pseudo-spectrum for non-selfadjoint differential operators.**Abstract**: We review some recent results by E.B. Davies et al around the notion of pseudo-spectrum (roughly the domain in the complex plane where the resolvent is large). Then we explain some recent work in progress in collaboration with N. Dencker and M. Zworski about resolvent estimates and absence of spectrum near the boundary of ``the'' pseudo-spectrum.*Thursday, April 18 at 15:15 room S1*

**Analysseminariet****Khalid Koufany**, Nancy, France: The Hilbert and Riemannian metrics of a symmetric cone**Abstract**: Let & be a symmetric cone (ie. an open convex self-dual and homogeneous cone in an Euclidean vector space). We study the Riemannian metric of & and prove that some semigroup naturally associated with & deceases the compounds of this metric. We also study the Hilbert projective metric of & and give some applications of it. The two metrics are characterized using the theory of Euclidean Jordan algebras.*Tuesday, April 16 at 15:15 room S1*

**Analysseminariet****Olli Martio**, University of Helsinki: Analysis on metric spaces**Abstract**: Let (X,d) be a metric space with a Borel measure. The concept of the first order Sobolev space of functions f from an open set D of the euclidean space into reals is extended to the case where D = X and to the case where the real numbers is replaced by X. Applications to the calculus of variations and to the potential theory are considered.*Tuesday, April 9 at 15:15 room S1*

**Analysseminariet****Leonid Nizhnik**, Kiev: Schröger operator with a ´'-interaction on Cantor set**Abstract**: A one-dimensional Schröger operator,*L*, with a point ´'-interaction on a finite set*X={x*of points with intensities ² =(²_{1},...,x_{n}}_{1},..., ²_{n}) is defined by the differential expression*-d^2/dx^2*on functions Æ (x) that belong to the Sobolev space*W*and satisfy, in points of the set_{2}^{2}(R\ X)*X*, the following conjugation conditions:Æ'(x

_{k}+0)= Æ'(x_{k}-0), Æ(x_{k}+0) - Æ(x_{k}-0) = ²_{k}Æ'(x_{k}).Let all the intensities of ´'-interactions satisfy ²

_{k}<0 (k=1,..., n). Then the self-adjoint Schrödinger operator*L*has exactly*n*distinct eigenvalues on the negative semiaxis (new result).

The operator*L*defined for the case of a closed set*X ‚ R*, of Lebesgue measure zero, with a Borel measure.^{1}

The Schrödinger operator*L*on a Cantor set*X*, with a standard Hausdorff measure and ²<0 , has an infinite number of negative eigenvalues (new result).*Tuesday, April 2 at 15:15 room S1*

**Analysseminariet****Peter Sjögren**: Maximaloperatorer för Ornstein-Uhlenbeck-halvgruppen med komplex tidsparameter.**Abstract**: Det handlar om uppskattningar i L^{p}för dessa operatorer. Tidigare resultat av Garcia-Cuerva, Mauceri, Meda, Sjögren och Torrea gäller framför allt fallet p < 2. Vi skall nu se att situationen är ganska annorlunda för p > 2.*Tuesday, March 19 at 15:15 room S1*

**Analysseminariet****Peter Jones**, Yale University, New Haven: From Cauchy Integrals to High Dimensional Data Sets: The Search for Low Dimensional Structures**Abstract**:In many problems of classical analysis or applied mathematics a basic step is to locate low dimensional subsets with some additional geometric structure. A standard example is to find points have a tangent plane (with some suitable definition). Over the past 15 years has seen the development of an L^{2}theory, as opposed to the usual a.e. statements. We present in this talk some joint work with Gilad Lerman on theorems in geometric measure theory and related computational algorithms. The main theorem gives a sharp condition on the surface area of a d-dimensional "nice" surface that is needed in order to hit a large piece of the support of a probability measure in n-dimensional space. Estimates of this kind had their origins in the study of L^{2}estimates for singular integrals like the Cauchy Integral on curves. The theorem explains why certain elementary multiscale algorithms successfully locate low dimensional subsets of high dimensional data sets.*Friday, March 15 at 15:15 room S1*

**Analysseminarium****Sergey Naboko**, Saint Petersburg: Operator theory and the analytic Nevanlinna functions analysis.**Abstract**: We plan to consider a part of the theory of analytic operator-valued Nevanlinna functions in the upper half-plane in connection with various problems of operator theory,such as perturbation of spectra,structure of singular spectra,Volterra operators theory,etc.No preliminary knowledges in this field,except the basic facts of complex analysis and operator theory, are necessary.*Tuesday, March 12 at 15:15 room S1*

**Gemensamt statistik och analysseminarium****Jeff Steif**:Stationary Determinantal Processes.**Abstract**: Given a function f on the d-dimensional torus with values in the unit interval, there is a 2-state stationary random field on the d-dimensional integer lattice that is defined via minors of the d-dimensional Toeplitz matrix of the function f. (This will be explained in detail with no assumed background). The variety of such systems includes certain combinatorial models, certain finitely dependent models, and certain renewal processes in one dimension. Among the interesting properties of these processes, we focus mainly on whether they have a phase transition analogous to that which occurs in statistical mechanics. There are connections to other things such as Szego's limit theorem for Toeplitz matrices and polynomial approximation. [This is joint work with Russ Lyons.]*Tuesday, March 5 at 15:15 room S1*

**Analysseminariet****M.Melgaard, G.Rozenblioum**:Eigenvalue asymptotics for weakly perturbed Dirac operator in constant magnetic field.**Abstract**: Continuation of the talk given in December 2001, but essentially independent. The Dirac operator describes quantum relativistic particles. In the constant magnetic field, the spectrum of the Dirac operator consists of discrete eigenvalues with infinite multiplicity, these eigenvalues and corresponding eigenspaces closely related to the ones of the Schrödinger operator. We find the asymptotics of eigenvalues arising when the Dirac operator is perturbed by a weak electric potential, thus expanding the results obtained earlier for the Schrödinger operator. All basic information on the Dirac operator will be given.*Tuesday, February 19 at 15:15 room S1*

**Analysseminariet****Johannes Brasche**: Zero range interactions**Abstract**: Several classes of quantum mechanical Hamiltonians H, describing a zero range interaction, will be treated. In particular,- infinitesimal generators of a Brownian motion with killing,
- infinitesimal generators of a superposition of a Brownian motion and a diffusion process on a submanifold,
- Hamiltonians describing an interaction on a polar (w.r.t. Brownian motion) set.

A method of construction will be given and a condition which is sufficient in order that the generalized wave operators

*W*exist and are asymptotically complete.^{±}(H,-¿)*Tuesday, February 5 at 15:15 room S1*

**Analysseminariet****Kari Astala**, Jyväskylä: Elliptic equations of non-divergence type and quasiconformal mappings**Abstract:**The inequality of Alexandrov, Bakelman and Pucci is a basic tool in the study of elliptic equations of non-divergence type. The result is e.g. the starting point for proving smoothness, Harnack-type inequalities and existence results in the non-linear theory. The inequality says, roughly, that the corresponding Green's operator is bounded from L^{n}to L^{}.

In the sixties Pucci suggested the precise L^{p}-range, in terms of the ellipticity, for the boundedness of the Green's operator. In the talk we will describe a proof for this in the two dimensional case; the proof uses strongly the properties of planar quasiconformal mappings.*Tuesday, January 29 at 15:15 room S1*

**Analysseminariet****Jörg Schmeling**, Lund: Some applications of dynamical dimension theory**Abstract:**In this talk we want to show how several methods in the dimension theory of dynamical systems can help to understand or unify questions arising outside this theory. The first application is a general method to estimate sets of real numbers defined by properties of their expansions. These applications unify and extend classical work of Borel, Besicovitch, Eggleston, and Billingsley. Other examples are connected to the continued fraction algorithm. One is an improvement of Denjoy's Theorem on circle diffeomorphisms. The other is on strong recurrence in polygonal billards. We also will discuss the role of non-standard multifractal analysis to study very strong recurrence properties of some classes of random walks. This will help to study syndetic numbers, Hardy-Weierstrass functions and lacunary Taylor series.*Tuesday, January 22 at 15:15 room S1*

**Analysseminariet****O.A. Ziza**, Moskva: Summability of orthogonal series.**Abstract:**At the beginning of the talk we remind the definition of various methods, mainly classical, of summability of numerical series. Then we consider their appications to orthogonal series*c*, where_{n}f_{n}*c*and_{n}^{2}<*{f*is an orthogonal system in_{n}}*L*. Such a series converges in^{2}[0,1]*L*, but we are interesting in convergence or summability a.e. on [0,1].^{2}We will describe some known results and some problems in the following directions.

1. Comparison of summability methods.

2. Weyl multipliers for convergence and summability.

3. Estimates of the rate of convergence or summability.

*Tuesday, January 15 at 15:15 room S1*

**Analysseminariet****M.S. Agranovich**, Moskva: Spectral properties of some boundary value problems for second order strongly elliptic systems.**Abstract:**For a class of systems indicated in the title, we consider boundary value problems with spectral parameter in the system or in the first order boundary conditions. The domain is smooth (easy parts of the talk) or Lipschitz. The main question is to find out in which Sobolev spaces**H**of positive order^{t}**t**in the domain or on its boundary do the eigenfunctions form a basis. This question is suggested by some physical problems for the Schrödinger equation.*Tuesday, January 8 at 15:15 room S1*

**Analysseminariet****Peter Sjögren**: Randkonvergens för Poissonintegraler i bidisken och i symmetriska rum - en översikt.**Abstract:**Med olika varianter av Poissonkärnan kan man producera egenfunktioner till flera translationsinvarianta differentialoperatorer eller till enbart Laplace-Beltrami-operatorn. Efter normalisering konvergerar de i allmänhet mot randfunktionen då man närmar sig randen på rätt sätt. Konvergens nästan överallt visas med hjälp av maximalfunktioner.### Seminars in 2001

*Tuesday, December 11 at 15:15 room S1*

**Analysseminariet****Victor Shulman**, Vologda, Russia: Schur multipliers and**C**-algebras.^{*}**Abstract:**Let*f(x,y)*be a continuous function on*X× Y*, where*X*,*Y*are sigma-compact spaces. By definition,*f*is a Schur multiplier with respect to measures*m*,*n*on*X*,*Y*, if, for any nuclear integral operator*T: L*with the kernel^{2}(X,m) ’ L^{2}(Y,n)*t(x,y)*, the function*f(x,y)t(x,y)*is again the kernel of a nuclear operator. It is proved that if*f*is a Schur multiplier with respect to*m*,*n*then it is a Schur multiplier for any other measures with the same supports. Some non-continuous and non-commutative versions of this result are also discussed.*Tuesday, December 4 at 15:15 room S1*

**Analysseminariet****Vladimir Nazaykinskiy**, Moskva: Semiclassical Statements of Control Problems for the Schrödinger Equation**Abstract:**We consider the Schrödinger equation with Hamiltonian depending on a control variable and give semiclassical statements of the controllability and stabilization problems in this case. Physically meaningful feedback controls given by mean values of observables are used. It is shown how the global WKB (Maslov's canonical operator) helps one to reduce these problems to related control problems for the classical Hamiltonian system associated with the Schrödinger equation and for the Hamilton-Vlasov system.*Tuesday, November 20 at 15:15 room S1*

**Analysseminariet****Michael Kempe**: Riesz means for certain operators related to the metaplectic representation**Abstract:**Given a self-adjoint operator*A*on**L**, it is always possible to consider the so-called Riesz means of^{2}*A*, which are special spectral multipliers involving a real parameter. One important question is whether the Riesz means are uniformly bounded operators on**L**. In our case the^{p}*A*'s will be certain second order differential operators on**R**, related to some interesting operators on the Heisenberg group as well as to the metaplectic representation. We will also discuss the relationship between Riesz means and so-called restriction theorems which turns out to be very useful in proving boundedness results for Riesz means.^{2n}*Tuesday, November 13 at 15:15 room S1*

**Analysseminariet****Genkai Zhang**: Berezin transform on bounded symmetric domains.**Abstract:**We calculate the spectral symbol of the Berezin transform on real bounded symmetric domains and find the branching coefficients of holomorphic representations on complex bounded symmetric domains. As applications we give a unified approach to the hypergeomtric orthogonal polynomials of several variables related the classical root systems via the Berezin transform.*Tuesday, November 6 at 15:15 room S1*

**Analysseminariet****Philip Brenner**: On maximal and local decay of the energy of solutions of nonlinear Klein-Gordon equations**Abstract:**We will treat a class of nonliearities of powertype, where the resulting equation will be a pertubation in energy sense of the corresponding linear Klein-Gordon equation. Although the solutions of the linear and nonlinear equations are asymptotically equal in energy norm, there are interesting suggested differences in convergence rates, which will be discussed. I will also add a proof of the Strichartz inequality - much used not only in the present context.*Tuesday, October 16 at 15:15 room S1*

**Analysseminariet (joint with Statistiska semininariet)****Greg Lawler**, (Duke, Cornell and Mittag-Leffler): An introduction to the stochastic Loewner evolution**Abstract:**Oded Schramm introduced the stochastic Loewner evolution a couple of years ago as a candidate for conformally invariant processes. Recent results by Schramm, Wendelin Werner, and myself as well as results by Stanislav Smirnov have shown the power of this tool. This talk will be an introduction to SLE and some of it properties.*Tuesday, October 2 at 15:15 room S1*

**Analysseminariet****Anders Öhgren**: Calderon-Zygmund theory on the real affine group.**Abstract:**We discuss a recent paper by W. Hebisch and T. Steger, where they propose a version of Calderon-Zygmund theory suitable for groups of exponential growth. An interesting application is the**L**^{p}properties of some first order Riesz operators on the real affine group.*Tuesday, September 25 at 15:15 room S1*

**Analysseminariet****Torbjörn Lundh**: Geodesics on quotient--manifolds and their corresponding limit points.**Abstract:**We will discuss a problem about generalizing a well known result which gives a correspondence between returning geodesics on Riemann manifolds and conical limit points of a discrete group. We will also give an example of a Fuchsian group where the conical limit set is different from the part on the boundary where the so called archipelago of the discrete group is not minimally thin.*Tuesday, September 18 at 15:15 room S1*

**Analysseminariet****Martin Brundin**: Orlicz potentials with respect to powers of the Poisson kernel.**Abstract:**If the Poisson kernel of the unit disc is replaced with its square root, then normalised Poisson integrals of*L*^{p}boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Sjögern and Rönning. In this talk I will mention some recent results along these lines, in particular a convergence result with*L*^{p}replaced by a class of Orlicz spaces that resemble*L*^{p}. As a special case, we recover Rönning's result.*Tuesday, September 11 at 15:15 room S1*

**Analysseminariet****Toshio Horiuchi**, Ibaraki, Japan: Missing terms in Hardy's inequalities and related topics.**Abstract:**In this talk we shall study the classical Hardy's inequalities and improve them by finding out missing terms. As related topics we shall study blow-up solutions of a semilinear elliptic boundary value problem. We also improve weighted Hardy's inequalities, which will be fundamental to investigate singular solutions of the*p*-harmonic equation.*Tuesday, September 4 at 15:15 room S1*

**Analysseminariet****Silvia Verzeletti**, Milano: Semigroups of operators.**Abstract:**We give a presentation of the general theory of semigroups, and illustrate some examples, focusing on the Ornstein-Uhlenbeck semigroup. We introduce the concept of semigroups subordination of order 1/2 of the heat semigroup on**R**^{d}endowed with both Lebesgue and Gaussian measures.*Tuesday, August 30 at 10:15 room S1*

**Analysseminariet****Hiroaki Aikawa**, Shimane University: Positive harmonic functions of finite order in a Denjoy type domain**Abstract:**A domain whose complement is included in a hyperplane is called a Denjoy domain. Benedicks studied positive harmonic functions in a Denjoy domain vanishing on the boundary. After that there are many studies. Recently, Poggi-Corradini considered a domain whose complement lies in a strip and studied harmonic functions of finite order. We show that some of his results remain to hold for a domain whose complement is included in a set wider than a strip.*Tuesday, August 28 at 15:15 room S1*

**Analysseminariet****Jonathan Arazy**, Haifa:The Weyl calculus on rank-one real symmetric domains.**Abstract:**We study the Weyl calculus in the framework of real symmetric domains*D=G/K*of rank one. The Weyl transform*W'W*is a function of the*G*-invariant Laplacian*L*:*W'W=f(L)*. We calculate the function*f*by solving a second-order recurrence. The new phenomenon is that in addition to the Gamma factors, also the Gauss hypergeometric functions appear naturally. The lecture will focus on the simplest case when*D*is the unit ball in C^{n}. This is a joint work with H. Upmeier.*Tuesday, August 14 at 15:15 room S1*

**Analysseminariet****M. M. Malamud**,Donetsk, Ukraine:Deficiency indices and selfadjointness of Hamiltonian systems**Abstract:**The main purpose of this talk is to investigate the formal deficiency indices*N*_{±}*(I)*of a symmetric first order system*Jf'+Bf=»H f*on an interval*I*, where*I=***R**or*I=***R**_{±}. Here*J,B,H*are n x n matrix valued functions and the Hamiltonian*H*e 0 may be singular even everywhere. We obtain two results for such a system to have minimal numbers*N*_{±}(**R**)=0 (resp.*N*_{±}(**R**_{±})=n) and a criterion for their maximality*N*_{±}(**R**_{±})=2n. Some conditions for a canonical system to have intermediate numbers*N*_{±}(**R**_{±}) are presented, too. We also obtain a generalization of the well-known Titchmarsh-Sears theorem for second order Sturm-Liouville type equations*Py:=-d/dx(A(x)*(2)^{-1}dy/dx+Q(x)y) + Q^{*}(x)dy/dx+R(x)y=»H(x)y,where

*A, Q, R, H L*and_{loc}(**R**)*A(x)*is positive definite for all*x*and**R***H(x)e 0.**Tuesday, May 29 at 15:15 room S1*

**Analysseminariet**

**Grigori Rozenblioum**:Toeplitz representation of the pseudodifferential quantization on singular manifolds.**Abstract:**Having an algebra A, a quantization is a linear unital mapping`F`of`A`into the algebra of bounded operators in a Hilbert space H which is 'almost' algebra homomorphism:`r(a,b)= F(a)F(b)-F(ab)`is a compact operator. A quantization is called Toeplitz if it has the form`F(a)=Pf(a)`where`f`is an algebra homomorphism of`A`into the algebra of bounded operators in some Hilbert space`K`,`H`is a subspace of`K`and`P`is the orthogonal projection onto`H`. Finding a Toepliz representations for a given quantization of an algebra is an important problem in noncommutative geometry. For the pseudodifferential quantization of the algebra of smooth functions on the cospheric bundle of a compact manifold, such representation was found in 80-s by V. Guillemin, using methods of complex analysis. In the present paper we construct a Toeplitz representation for the pseudodifferential quantization of the natural symbolic algebra on the manifold with cone-like singularities.*Tuesday, May 22 at 15:15 room S1*

**Analysseminariet**

**Jan-Olav Rönning**, Inst. för Naturvetenskap, Högskolan i Skövde :Fractal dimensions of product sets in infinite product spaces**Abstract:**We consider the infinite dimensional complete metric space**R**^{}with the product topology and Frechet metric d(x,y)=\sum 2^{-n}min{1,|y_{n}- x_{n}|}, which is a natural infinite dimensional extension of the n-dimensional Euclidian spaces. In this setting we consider product sets \prod E_{n}, where the coordinate set E_{n}typically is a fractal set, like for instance a uniform Cantor set, and investigate the relations between the Hausdorff (and box) dimensions of the product set and corresponding coordinate sets. In particular, we will indicate that the usual formula for finite dimensional products (which hold for at least reasonably "nice" fractal sets); dim(\prod E_{n})=\sum dim(E_{n}), does not usually hold in this setting, even for very simple sets. We will, however, state the proper replacement for this formula in our setting. We have also a result stating that for an infinite product of self-similar coordinate sets, the ordinary formula for finite dimensional product does still holds, if we replace the Frechet metric above with a more general one which, however, depends on the sets involved. Finally, we will give some formulas for the Hausdorff dimension of infinite products of uniform Cantor sets when we have these more general Frechet metrics.This talk describes a joint work with Professor Kathryn E. Hare, University of Waterloo, currently visiting Matematiska institutionen, CTH & GU.

*Tuesday, April 24 at 15:15 room S1*

**Analysseminariet**

**Johannes Sjöstrand**:Asymptotic distribution of eigen-frequencies for damped wave equations**Abstract:**The eigen-frequencies associated to a damped wave-equation are known to belong to a band parallel to the real axis and it follows from a general result of Markus and Matsaev, that their real parts are distributed according the standard Weyl law. In this talk we explain that up to a set of density 0, they are confined to a smaller band, determined by the Birkhoff limits of the damping coefficients, and that certain averages of the imaginary parts converge to the average of the damping coefficient.*Tuesday, April 11 at 15:15 room S1*

**Analysseminariet**

**Alexei Iantchenko**, Malmö:Asymptotic behavior of the one-particle density matrix of atoms at distances Z^{-1}from the nucleus**Abstract:**We prove that the suitably rescaled density matrix of ground states of atomic Schrödinger operators with nuclear charge Z converges on the scale 1/Z to the projection of the negative spectral subspace of the Schrödinger operator of the hydrogen atom (Z=1).*Tuesday April 4 at 15:15 room S1*

**Analysseminariet**

**Nico Spronk**, Waterloo, Canada: Operator Space Structure on the Fourier Algebra and Amenability Theory**Abstract:**Let**G**be a locally compact group,**L**be the^{1}(G)*group algebra*, and**A(G)**be the*Fourier algebra*of**G**. If**G**is Abelian,**A(G)<-> L**via the Fourier transform, where^{1}(G^)**G^**is the dual group. However, for non-Abelian groups it is more difficult to define. Never-the-less, it is still a commutative semi-simple Banach algebra with Gel'fand spectrum**G**, and holds many of the properties one might like to have for**L**, even though we may not have a reasonable candidate for ``^{1}(G^)**G^**''.

B. Johnson showed that**L**is^{1}(G)*amenable*(i.e. every derivation of**L**into a dual Banach^{1}(G)**L**-module is inner) precisely when^{1}(G)**G**is amenable (i.e. ``averagable'') as a group. It was long suspected the same might hold for**A(G)**too. However, Johnson showed that there exist compact groups $G$ for which**A(G)**is not amenable. The theory of operator spaces, developed in the last two decades, however, gives us a context in which we can say ``**A(G)**is amenable if and only if**G**is amenable''.*Tuesday, Mars 20 at 15:15 room S1*

**Analysseminariet**

**Vladimir Nazaikinskii**, Inst for Problems in Mechanics, Moskava: Noncommutative analysis: theory and applications**Abstract:**Noncommutative analysis deals with functions of several noncommuting operators and has numerous applications to algebraic problems, differential equations, asymptotics, etc. A concise survey of basic elements of the theory will be given along with a variety of specific applications, including the Campbell-Hausdorff-Dynkin formula, the Poincare-Birkoff-Witt theorem, and some other problems.*Tuesday, Mars 13 at 15:15 room S1*

**Analysseminariet**

**Peter Sjögren**:Some Riesz operators on the affine group and other solvable Lie groups.**Abstract:**If*L*denotes the Laplacian on the group, Riesz operators are typically*XL*and^{-1/2}*L*, where^{-1/2}X*X*is an invariant vector field. The question is whether they are bounded on*L*. The main difficulty is the global behaviour of these operators.^{p}*Tuesday, Mars 6 at 15:15 room S1*

**Analysseminariet**

**Philip Brenner**:Truncated Besov space norms**Abstract:**In applications e g to PDE it is of interest to be able to describe the spaces that appear by interpolation between the set of functions**u(x,t)**on**R**which for some^{n}R**T > 0**belong to**H**on^{s}(R^{n})**(0,T)**and the space of functions which belong to**H**on a fixed interval^{1}**(0,T**. The interpolation space can be described by a set of semi-norms, which we for obvious reasons will call truncated Besov norms. We will show that these norms share some important properties and characterizations with the usual norms on Besov spaces. I will also indicate some applications. This is joint work with Peter Kumlin._{0})*Tuesday, February 27 at 15:15 room S1*

**Analysseminariet**

**Stefan Böcker**,Bochum: On random Schrödinger operators with Poisson potential and survival asymptotics for Brownian motion**Abstract:**With the aid of Tauberian theorems it can be shown that the expansion of the integrated density of states at the infimum of the spectrum can be calculated via the survival asymptotics for a Brownian motion. This known fact will be explained in some detail. Moreover (new) results on the survival asymptotics for a Brownian motion in a scaled Poissonian potential will be given.*Tuesday January 30 at 15:15 room S1*

**Analysseminariet**

**Elizaveta Zeldina**, St.Petersburg: Bergman Kernels for C^{6}and H^{5}-Smooth Almost Spherical Domains

**Abstract:**It is a joint work with professor N.Shirokov. Let**D c C**be an almost spherical domain with C^{n}^{6}-smooth boundary. Under some unessential restrictions on**D**it is possible to find first two terms and the remainder term for the asymptotics of the Bergman kernel**B**as_{D}(z, z)**z**tends to the boundary. These terms have the same rate of growth as the corresponding terms for the C^{}-smooth strictly pseudoconvex domains. Further on, it turns out that lowering of the smothness form C^{6}to H^{5}implies the presence of an additional term that is absent in C^{}case.*Tuesday, January 23 at 15:15 room S1*

**Analysseminariet**

**Johannes Brasche**: Some convergence results for Schrödinger operators and Schrödingers equation.**Abstract:**In special cases the solution of the Schrödinger equation can be approximated by finite sums of stationary solutions locally uniformly in time. Upper estimates for the rate of convergence will be given.

Also a result on one - dimensional Schrödinger operators with signed - measure valued potentials will be given. If the potentials have finite total charge and converge weakly then the corresponding Schrödinger operators converge in the norm resolvent sense. Upper estimates for the rate of convergence will be given. Via a combination with the "main theorem of statistics" (Glivenko - Cantelli) one gets convergence results for certain sequences of random Schrödinger operators.

### Seminars in 2000

*Wednesday December 20 at 15:15 room S1*

**Analysseminariet**

**Martin Brundin**: Approach regions for the square root of the Poisson kernel of the unit disk.**Abstract:**If the Poisson kernel of the unit disc is replaced by its square root, it is known that normalised Poisson integrals of L^{1}boundary functions converge almost everywhere at the boundary, along approach regions wider than the ordinary nontangential cones. Convergence results are known also for the subspaces L^{p}, 1<p d , and weak L^{p}, 1< p< . In this talk I will explain how questions of convergence easily and naturally are reduced to proving suitable estimates for corresponding maximal operators, and how this can be done.*Tuesday December 12 at 15:15 room S1*

**Analysseminariet**

**Philip Brenner**: On the decay of solutions to nonlinear Klein-Gordon equations.**Abstract:**Assuming only finite energy ( not necessarily small) data, what can be said about the decay of solutions to nonlinear Klein-Gordon equations with power-type nonlinearities? In view of the Strichartz estimates ( time-space estimates ) for the linear equation, solutions of that equation decays in the mean for large times even for finite energy data - but the rate of decay is usually not known. I will give some new estimates for solutions of the nonlinear equation, which in a sense are optimal in the case of maximal decay.*Tuesday December 5 at 15:15 room S1*

**Analysseminariet****Victor Shulman**, Vologda: On Invariant Subspaces for Operator Semigroups and Operator Lie Algebras.**Abstract:**The notion of the joint spectral radius for a system of operators (introduced by Rota) has recently got many applications to Invariant Subspace Theory. In this talk we will discuss the connection and the related techniques. In particular, using the spectral radii technique we will show that- Any Lie algebra of compact quasinilpotent operators has an invariant subspace;
- Any semigroup of compact quasinilpotent operators has an invariant subspace (Turovskii's solution of the Volterra Semigroup Problem);
- Any semigroup of operators, whose spectral radii is equal
to essential spectral radii, has an invariant subspace provided
it contains a group Exp(
*tV*):*t*->**R**, where*V*is a non-zero compact operator.

**Lectures about Representation Theory**

Our visiting professor Kathryn Hare will give a series of 4 talks on

*Representations of compact Lie groups with applications to harmonic analysis.*

Start: Tuesday, November 7 at 15.15 in S1.

**Abstract**The goal of this series of talks will be to introduce graduate students to the representation theory of compact Lie groups from the perspective of an analyst, and give applications in harmonic analysis. We will focus mainly on concrete examples to minimize technicalities. No prior knowledge of Lie theory will be assumed. I would like to sketch proofs of the Weyl character and dimension formulas, and illustrate their use in research problems.*Tuesday October 31 at 15:15 room S1*

**Analysseminariet****Vasyl Ostrovskyi**, Kiev: Centered one-parameter semigroups and representations of double commutator relations

**Abstract:**In 1974, Morrel and Muhly introduced the class of centered operators, which arise naturally in representations of certain classes of***-algebras. We extend this notion to the case of centered one-parameter semigroups, and study their properties. In particular, we establish the Wold decomposition for such semigroups and give complete description of one-parameter centered semigroups of partial isometries. It is shown that to a centered one-parameter semigroup there corresponds a natural (unbounded) representation of the relation*[A,[A,B]]=0*with self-adjoint*A*, and*B*being a closed extension of a symmetric operator. This correspondence is similar to the correspondence between representations of a Lie algebra and representations of the corresponding Lie group.*Tuesday October 10,17 at 15:15 room S1*

**Analysseminariet****Lyudmila Turowska**: Operator synthesis, tensor algebras and harmonic analysis.

**Abstract:**In this talk we shall discuss the notion of operator synthesis (introduced by W.Arveson) and its application and connection with tensor algebras and the notion of spectral synthesis in harmonic analysis. In particular, we shall explore the question on equivalence of synthesis with respect to the Varopoulos algebra*V(X,Y)=C(X) C(Y)*and operator synthesis. The talk will contain necessary information and examples from this field and should be understandable for a wide audience.*Tuesday October 3 at 15:15 room S1*

**Analysseminariet****Rolf Liljendahl**: The maximal operator for some non-doubling measures.

**Abstract:**We will consider the maximal operator with respect to non-centered balls and a non-doubling measure. Results on the boundedness will be given for some simple measures with densities depending only on the first coordinate. The main ideas of the proof will also be discussed.*Tuesday September 26 at 15:15 room S1*

**Analysseminariet****Michael Melgaard**: The Schrödinger equation near thresholds.

**Abstract:**A number of closely related problems are tackled concerning the study of the threshold behaviour of resolvents of Schrödinger-type 2x2 operator-valued matrix Hamiltonians. Results are given in two directions:

**1. Perturbation of eigenvalues and half-bound states (zero resonance) embedded at a threshold**

2. Asymptotic expansions of the resolvent as the spectral parameter tends to a threshold.

Applications are given to the Friedrichs' model, a quark model and scattering theory for pairs of two-channel Hamiltonians with Schrödinger operators as component Hamiltonians.*Tuesday September 12 at 15:15 room S1*

**Analysseminariet****Kathryn Hare**: The Littlewood-Paley Theorem

**Abstract:**The classical Littlewood-Paley Theorem and its generalizations have been an important topic in harmonic analysis for many years. It is natural to ask if there are partitions other than (the known examples of) lacunary sets and their iterates for which such a theorem holds. We will report on joint work with Klemes which partially answers the conjecture that the Littlewood-Paley theorem holds for any rearrangement of a lacunary partition. Our methods give a new and more elementary proof of the classical theorem and easily transfer to other settings.*Tuesday September 5 at 15:15 room S1*

**Analysseminariet****Grigori Rozenblioum**: Trace class pseudodifferential calculus and unusual index formulas.

**Abstract:**Some formulas in analysis make sense under more restric\-ting conditions than the existence of the object the formulas describe. For example, the expression

*(2 i)*for the winding number of the curve in the complex plane described by a function^{-1}_{}(a'a^{-1})dx*a(x)*on the circle, makes sense only under some smoothness conditions for*a*while the winding number itself exists for a continuous*a*. A similar situation arises when one tries to find a formula for the index of pseudodifferential operators with operator-valued symbols (such operators are very important in analysis on singular manifolds). The usual index formulas lose sense, and one has to find new ones. We are going to describe how the calculus of pseudodifferential operators with operator-valued symbols is developed (the usual calculus will be explained as well) and to show how algebraical methods produce new index formulas.*Tuesday August 29 at 15:15 room S1*

**Analysseminariet****Jonathan Arazy**, Haifa: Invariant symbolic calculi and eigenvalues of link transforms on symmetric domains.

**Abstract:**In this talk I will survey a recent joint work with H. Upmeier in which we study the structure of the invariant symbolic calculi and establish a new method to compute the eigenvalues of the link transforms. In the first part of the talk I will describe the main results, and in the second part I will give some details of the work in the context of the Fock space.*Tuesday May 23 at 15:15 room S1*

**Analysseminariet****Vladimir Kapustin**Reflexivity of contractions close to isometries

**Abstract:**We prove a criterion for the reflexivity of an algebra of operators generated by a single contraction*T*on a Hilbert space. An algebra*A*of operators is said to be reflexive if every operator preserving the lattice of invariant subspaces of*A*belongs to*A*. The most important case is that of scalar inner characteristic function, where*T*is a contractive one-dimensional perturbation of a singular unitary operator. In this case, the problem of reflexivity can be rewritten and solved in terms of theory of fuctions in the unit disk.*Tuesday May 16 at 15:45 room S1*

**Analysseminariet och CAM-seminariet****Margaret Cheney**, Rensselaer Polytechnic Institute and LTH: Optimal Acoustic Measurements

**Abstract:**We consider the problem of obtaining information about an inaccessible half-space from acoustic measurements made in the accessible half-space. If the measurements are of limited precision, some scatterers will be undetectable because their scattered fields are below the precision of the measuring instrument. How can we make optimal measurements? In other words, what incident fields should we apply that will result in the biggest measurements? There are many ways to formulate this question, depending on the measuring instruments. In this talk we consider a formulation involving wave-splitting in the accessible half-space: what downgoing wave will result in an upgoing wave of greatest total energy? A closely related question arises in the case when we have a guess about the configuration of the inaccessible half-space. What measurements should we make to determine whether our guess is accurate? In this case we compare the scattered field to the field computed from the guessed configuration. Again we look for the incident field that results in the greatest energy difference. We show that the optimal incident field can be found by an iterative process involving time reversal ``mirrors''. For band-limited incident fields and compactly supported scatterers, this iterative process converges to a finite sum of time-harmonic fields. In other words, the optimal incident field is generaly time-harmonic. This provides a theoretical foundation for the pulse-broadening observed in certain computations and time-reversal experiments. Moreover, this result suggests that from the point of view of distinguishing the presence of a scatterer, the chirps and pulses that are usually used may not be best.*Tuesday May 9 at 15:45 room S1*

**Analysseminariet****Fausto Di Biase**, Sassari: On the McMillan Twist Theorem

**Abstract:**Further progress toward a new proof of the McMillan Twist Theorem (joint work with N.Arcozzi, E.Casadio-Tarabusi and M.Picardello).*Tuesday May 2 at 15:45 room S1*

**Analysseminariet****Peter Sjögren**: Om maximaloperatorn för Ornstein-Uhlenbeck-semigruppen.

**Abstract:**Vi skall bl.a. presentera ett nytt bevis för att denna operator är av svag typ 1,1 i ändlig dimension. Här är tidsparametern positiv, men även fallet med komplexvärd parameter skall beröras.*Tuesday April 25 at 15:15 room S1*

**Analysseminariet****Xavier Tolsa**: A discrete version of Calderon reproducing formula for homogeneous and non homogeneous spaces.

**Abstract:**The reproducing formula of Calderon is a very important tool in Harmonic Analysis, with applications in Littlewood-Paley theory and Calderon-Zygmund theory. The use of the Fourier transform is fundamental in the obtention of this formula. In the setting of homogeneous spaces the Fourier transform is not available. However in this case, following an idea of R. Coifman, one can obtain a Calderon type reproducing formula using the Lemma of Cotlar-Stein. In this talk I will explain this idea of R. Coifman, and I will show that his construction can also be extended to the case of non doubling measures.*Tuesday April 11 at 15:15 room S1*

**Analysseminariet****Nikolay Shirokov**, St.Petersburg: Constructive description of Hölder classes on the half - axis.

**Abstract:**Let H^{a}be the usual Hölder class of order**a**of functions defined on the half-axis [0,). We discuss the problem of description of the class H^{a}in terms of possible approximation rate for the functions in a proper nonuniform scale by entire functions. This situation may be compared with the classical Jackson-Bernstein theorem concerning the description of the Hölder class of 2pi-periodic functions by means of approximation by trigonometric polynomials.*Tuesday April 4 at 15:15 room S1*

**Analysseminariet****Nikolay Shirokov**, St.Petersburg: Possible rate of decay of a (P,A)-lacunary function

**Abstract:**Let , , be a function analytic in the unit disc. We call the function f (or the corresponding power series) (P,A)-lacunary for p>1 and A>0, if there exists a subset of natural numbers such that a_{n}=0 if and . It turns out that f(x) cannot decay arbitrarily fast as x->1-0. The optimal rate of decrease of a (P,A)-lacunary function will be discussed.*Tuesday Mars 28 at 15:15 room S1*

**Analysseminariet****Fernando Soria**, Madrid: On a dynamical system related to estimating the best constant in an inequality of Hardy and Littlewood.

**Abstract:**On a dynamical system related to estimating the best constant in an inequality of Hardy and Littlewood: It is still unknown whether or not the best constant, C_{n}, in the weak type 1 inequality of the centered Hardy-Littlewood maximal operator is uniformly bounded in the dimension n. In this work we make an attempt to determine the best constant in dimension 1.*Tuesday Mars 21 at 15:15 room S1*

**Analysseminariet****Nicola Garofalo**, Purdue University and Inst. Mittag-Leffler: Properties of harmonic measures in the Dirichlet problem for Lie groups of Heisenberg type.

**Abstract:**In the theory of boundary value problems for second order PDEs the study of harmonic measure occupies a central position. In 1977 B. Dahlberg established his famous result stating that on a Lipschitz domain harmonic and surface measure are mutually absolutely continuous. Furthermore, the Radon-Nikodym derivative satisfies a reverse Hölder inequality at every scale. A basic consequence of the latter is the solvability of the Dirichlet problem when the boundary datum belongs to a Lebesgue space with respect to surface measure.*Tuesday Mars 7 at 15:15 room S1*

**Analysseminariet****Torbjörn Lundh:**Martin Boundary of a Fractal Domain.

**Abstract:**We determine the Martin boundary, using the boundary Harnack principle, of the complement to self-similar fractals in a certain class. A joint work in progress with H. Aikawa.*Tuesday February 29 at 15:15 room S1*

**Analysseminariet****Vasily Vasyunin, St.Petersburg:**The characteristic function of a contraction and its applications

**Abstract:**I will discuss the aims of models for operators in general, and explain the main steps of construction of the coordinate-free function model for Hilbert space contractions. It generalizes the models proposed by Szokefalvy-Nagy - Foias, de Branges - Rovnyak, and some others. The basic object in this theory is the so-called characteristic function of the contraction under investigation. This function determines the initial contraction up to unitary equivalence, and in these terms all results of the model theory are formulated. After introducing basic objects I will consider two questions: how to describe in terms of characteristic function the spectrum of a contraction and the lattice of its invariant subspaces. At the end I give two examples of operators, for which the characteristic function will be calculated. Namely, these will be the operator of integration and the Schrödinger equation on the half-line with complex boundary condition.*Tuesday February 1,22 at 15:15 room S1*

**Analysseminariet****Johannes Brasche:**Inverse Spectral Theory for Self-adjoint Extensions.

**Abstract:**In various models in quantum mechanics the information about the Hamiltonian*H*of the system is incomplete in the following sense: One is given a symmetric operator*S*which has infinitely many self-adjoint extensions and one only knows that*H*is one of these extensions. This is one of the strong motivations to investigate the following general question: What kinds of spectra can the self-adjoint extensions of a given symmetric operator have and how can one find self-adjoint extensions with preassigned kinds of spectra?

In the talk I shall explain what is meant by `kinds of spectra' and give partial answers to the above question.*Tuesday February 8 at 15:15 room S1*

**Analysseminariet****Hiroaki Aikawa,Shimane University och Inst. Mittag-Leffler:**Capacity estimate and tangental Nagel-Stein type theorem.

**Abstract:**We estimate the capacity of sets under a certain enlargement operation and give an application to the boundary behavior of Nagel-Stein type.*Tuesday January 25 at 15:15 room S1*

**Analysseminariet****Sergey Kislyakov, St.Petersburg:**Projections and partial retractions for weighted Hardy spaces.

**Abstract:**The Riesz projection (i.e., the orthogonal projection of L^{2}onto H^{2}) maps L^{p}onto H^{p}for 1, has weak type (1,1), and is discontinuous on L

^{1}and L^{}. It will be discussed what can substitute this operator in the case of weighted spaces with a general weight, and in the case of extreme indeces 1 and .*Monday January 24 at 15:15 room S1*

**Analysseminariet****Tao Qian, University of New England, Australia:**Fourier analysis on the unit sphere of**R**^{n}.

**Abstract:**We wish to show that the complex structure for**R**^{n}using Clifford algebra is the appropriate one for studying Fourier analysis on the sphere. The talk will concern the basic topics such as Fourier series, Fourier multipliers, singular integrals, including the Cauchy integral on the sphere, and will introduce Clifford analysis in relation to the context. We will introduce Fueter's and Sce's devices, and the generalisation of the speaker of inducing Clifford holomorphic functions from holomorphic functions of one complex variable, and applications.

**Analysseminariet***Tuesday January 11 at 15:15 room S1*

**Analysseminariet**

**Alexei Alexandrov, S:t Petersburg:**Remarks concerning imbedding theorems for co-invariant subspaces of the shift operator.

**Abstract:**Let*O*be an inner function. With this function we associate a subspace*O*of the Hardy space^{*}(H^{p})*H*. We investigate measures on the unit disc^{p}*D*such that*O*c^{*}(H^{p})*L*.^{p}(µ)*Tuesday January 11 at 15:15 room S1*

**Analysseminariet**

**Lydmila Turowska**:Tame and wild problems in the theory of *-representations.

**Abstract:**This is an introductory lecture on representation theory of *-algebras and, in particular, the complexity of the description of *-representations by bounded and unbounded operators. I am going to discuss the notion of *-wild algebras for which the classification of representations includes the classical unsolved problem on the canonical form of one non-selfadjoint operator (or two non-commuting selfadjoint operators) in a Hilbert space. No special knowledge of the theory of *-algebras and C^{*}-algebras is required.*Tuesday December 14 at 15:15 room S1*

**Analysseminariet**

**Michal Wojciechowski, Polish Academy of Science**: The rational Fourier multipliers arising in the theory of anisotropic Sobolev spaces.*Tuesday November 30 at 15:15 room S1*

**Analysseminariet**

**Genkai Zhang**: Nearly holomorphic functions and discrete spectrum of invariant differential operators.

**Abstract:**We consider the discrete spectra of invariant differential operators on a weighted L^{2}-space on a bounded symmetric domain. The first discrete spectrum corresponds to the weighted Bergman space of holomorphic functions. We prove that the functions in the other eigenspaces are nearly holomorphic functions in the sense of Shimura.*Tuesday November 23 at 15:15 room S1*

**Analysseminariet**

**Grigori Rozenblioum**: Standard and non-standard traces - Part 3.

**Abstract:**In the final part of the series the relations between two traces on pseudodifferential operators, spectral properties and poles of Zeta-functions will be studied and we finish with the Atiyah-Singer theorem.*Tuesday November 16 at 15:15 room S1*

**Analysseminariet**

**Grigori Rozenblioum**: Standard and non-standard traces - Part 2.

**Abstract:**We will study non-regular traces of operators which will lead us to traces of pseudodifferential operators. The relation with the zeta-function will be explained. The third part, 23rd November, will deal with the Atiyah-Singer index theorem.*Tuesday November 9 at 15:15 room S1*

**Analysseminariet**

**Grigori Rozenblioum**: Standard and non-standard traces and some applications.

**Abstract:**When one tries to carry over the trace of matrices to operators some other algebras, certain complications arise. This leads to the notion of non-standart, or singular traces. In a series of two lectures I'll try to explain how these traces can be described. The applications will, in particular, lead to a very easy formulation and proof of the Atiyah-Singer index theorem.*Tuesday November 2 at 15:15 room S1*

**Analysseminariet**

**Xavier Tolsa**: Calderon-Zygmund theory for non doubling measures.

**Abstract:**Usually, in order to study Calderon-Zygmund operators in the spaces L^{p}(µ), it is assumed that the measure µ is doubling. This is an essential assumption in the classical Calderon-Zygmund theory. However, recently it has been shown that many results in this theory also hold without the doubling assumption. In this talk we will explain some of the differences and difficulties that arise when one works with non doubling measures, and how they have been solved to obtain some of the results mentioned above.*Tuesday October 26 at 15:15 room S1*

**Analysseminariet**

**Lennart Frennemo**: Generella Tauberproblem i en och flera dimensioner.

**Abstract:**Från Wieners allmänna Taubersats till n-dimensionella Tauberproblem i viktade rum. Tillämpat på t.ex. multidimensionella Laplacetransformer blir resultaten skarpa.*Tuesday October 26 at 15:15 room S1*

**Analysseminariet**

**Prof. Giancarlo Mauceri, Genua, hedersdoktor vid Göteborgs universitet**: Functional calculus for the Ornstein-Uhlenbeck operator.

**Abstract:**Consider a self-adjoint operator*A*on L^{2}for some measure space. If*m*is a bounded Borel function on**R**, one can define a bounded operator*m(A)*on L^{2}by means of the spectral resolution of*A*. For 1Ø p< , the function*m*is called an L^{p}*-multiplier*if*m(A)*extends to a bounded operator on L^{p}. These multipliers form a Banach algebra. Necessary or sufficient conditions for L^{p}-multipliers have useful applications in spectral theory, in potential theory, to partial differential equations, in scattering theory ...

In the last thirty-odd years this problem has been investigated for several generalized Laplacians ( Laplace-Beltrami operators on Riemannian manifolds, sums of squares of vector fields, Schrödinger operators etc.)

I shall discuss some recent results for the Ornstein-Uhlenbeck operator, a "natural" Laplacian on the Euclidean space with Gauss measure. The talk will be nontechnical, aimed at a general audience of analysts.*Tuesday October 12 at 15:15 room S1*

**Analysseminariet**

**Maria Roginskaya**: Två anmärkningar till tidigare föredrag.

**Abstract:**Först vill jag göra en anmärkning till Philippe Jamings föredrag den 7/9 och visa ett samband mellan ett av hans resultat och ett på 70-talet väl utforskat PDE-problem.

I den andra delen av seminariet vill jag, utan något samband med den första frågan, berätta om de framsteg som Michal Wojciechowski och jag gjort i den riktning som jag redogjorde för på ett seminarium den 6/10 1998.*Tuesday September 28 at 15:15 room S1*

**Analysseminariet**

**Peter Sjögren**: En uppskattning för en maximaloperator relaterad till Gaussmåttet.

**Abstract:**Om man i det Euklidiska rummet ersätter Lebesguemåttet med ett Gaussmått, kan man definiera en maximaloperator av Hardy-Littlewoods typ, med användning av ocentrerade klot. Vi skall visa att denna operator är begränsad på L^{p}m.a.p. Gaussmåttet.*Tuesday September 21 at 15:15 room S1*

**Analysseminariet****Boris Fedosov**, Moskva och Berlin: Pseudodifferential operators and deformation quantization.**Abstract:**First we consider a kind of coordinate-free description of pseudodifferential operators. The operator on a manifold X is represented as a flat section of some bundle over a phase space (that is over a cotangent bundle of X) with respect to a special connection. It turns out further that such objects make sense for general symplectic manifolds, not only for cotangent bundles. Such objects form an algebra which may be considered as a deformation of the algebra of functions. Like the algebra of psedodifferential operators it has a trace which allows to define the notion of index. The index theorem generalizing the famous Atiyah-Singer theorem holds for this algebra. It gives universal quantization conditions.*Tuesday September 14 at 15:15 room S1*

**Analysseminariet****Xavier Tolsa**: BMO, H^{1}and Calderon-Zygmund operators for no doubling measures.Abstract: It has been shown recently that many results of the classical Caderon-Zygmund theory also hold for non doubling measures. For instance, non doubling versions of the T1 and Tb theorems have been obtained. In fact, an application of a theorem of Tb type for non doubling measures has been essential for the recent solution of Vitushkin's conjecture for sets with finite length by G. David. In this talk we will introduce good substitutes for the spaces BMO and H

^{1}suitable for non doubling measures. We will show how several classical results involving BMO and H^{1}hold for their non doubling substitutes. For instance, we will see how the T1 theorem for the Cauchy transform for non doubling measures can be obtained by an interpolation theorem.*Tuesday September 7 at 15:15 room S1*

**Analysseminariet****Philippe Jaming**, Orléans: Hardy spaces on the real hyperbolic ballAbstract: The real hyperbolic ball is the euclidean ball

*B*_{n}endowed with the hyperbolic geometry. Let*S*^{n-1}be the unit sphere in*R*^{n}. Call*D*the associated Laplacian and say that a function*u*on*B*_{n}is -harmonic if*Du*=0.Define the Hardy space as

We will show that this space admits an atomic decomposition as well as characterisations in terms of non-tangential maximal functions and area integrals, thus generalising results of Garnett-Latter and Feffermann-Stein to the hyperbolic case.

*Tuesday August 31 at 15:15 room S1*

**Analysseminariet****Fausto Di Biase, Sassari.**Twist points in higher dimension.

**Abstract.**The Twist Point Theorem for planar domains asserts that almost every boundary point, with respect to harmonic measure, is either a twist point or a regular point. In this work in progress, in collaboration with Bert Fischer, we study a special domain in 3-space in order to understand and possibly extend the Twist Point Theorem to higher dimensions.*Tuesday August 24 at 13:15 room S1*

**Analysseminariet****Philippe Jaming, Orleans.**On some phase retrieval problems

**Abstract.**Phase retrieval problems from a large class of problems that appear in many different areas of physics such as chrystalography signal processing or quantum mechanics. So far, they have not attracted the attention they deserve in the mathematical comunity.*Tuesday May 24 at 15:15 room S1*

**Analysseminariet****Grigori Rozenblioum.**Homotopies in glued local C^{*}-algebras and index for pseudodifferential operators on singular manifolds.

For C^{*}-algebras generated by pseudodifferential operators on singular manifolds, the symbol, unlike the regular case, consists of several components, one of them responsible for the smooth part of the manifold, and the others living on the singular parts. These partial symbols, each of them belonging to a certain C^{*}-algebra, must be consistent with each other. Generalisation of this consistency condition leads to the notion of glued local C^{*}-algebras. The homotopy lifting theorem, which will be discussed in the talk enables one to lift the homotopy of invertible elements in one component of the glued algebra to a homotopy of invertible elements in the whole glued algebra. As an application, index formulas for a class of operators on manifolds with edges are obtained.*Tuesday May 18 at 15:15 room S1*

**Analysseminariet****Klaus Schmidt, University of Vienne.**De Finetti's Theorem revisited.

De Finetti's theorem states (in essence) that every probability measure on a full shift with finite state space which is invariant and ergodic under the action of the permutation group is i.i.d. Bernoulli. This result can be generalized significantly by using ergodic theory and leads to a variety of results on exchangeable events, the tail behaviour of stationary stochastic processes, or on the ergodicity of infinite covers of horocycle flows.*Tuesday May 11 at 13:15-15:00 room S1*

**Analysseminariet****Håkan Hedenmalm, Lund**Carlesons L^{2}-sats för Dirichletserier*Tuesday May 4 at 15:15-17:00 room S1*

**Analysseminariet****Johan Råde**Introduction to C^{*}-algebras (continuation)

Today I will discuss closed ideals and homomorphisms of C^{*}-algebras, and also C^{*}-algebras of pseudodifferential operators.*Tuesday April 23 at 15:15-17:00 room S1*

**Analysseminariet****Johan Råde**Introduction to C^{*}-algebras.

**Abstract:**This is the first of a series of two or three seminars where I will give a quick introduction to C^{*}-algebras. I will discuss: the definition of a C^{*}-algebra, commutative C^{*}-algebras, closed ideals and homomorphisms. I will describe several examples, in particular the C^{*}-algebra of order 0 pseudodifferential operators on a compact differentiable manifold. There will of couse not be time for proofs, that would require a one semester course, but I will describe the main ideas.*Tuesday Mar 30 at 15:15-17:00 room S1*

**Analysseminariet****Jose Maria Martell**Weights and vector-valued inequalities on nonhomogeneous spaces.

**Abstract:**In this talk I would like to explain some known facts about weights and vector-valued inequalities. With these results, some conditions on weights can be obtained for classical operators. I will follow the book by J. Garcia-Cuerva and J.L. Rubio de Francia: Weighted Norm Inequalities and Related Topics, North Holland, Amsterdam, 1985.*Tuesday Mar 23 at 15:15-17:00 room S1*

**Analysseminariet****Jose Maria Martell**Weights and vector-valued inequalities on nonhomogeneous spaces.

**Abstract:**Recently, some results about Calderon-Zygmund singular integral operators on nonhomogeneous spaces have been proved. A nonhomogeneous space is a metric space endowed with a non-negative measure that is not assumed to satisfy any doubling condition. The aim of the talk is to handle vector-valued Calderon-Zygmund operators on these spaces and use them to get certain types of weighted inequalities. Finally, the Cauchy integral operator will provide an interesting example for this theory.*Tuesday Mar 16 at 15:15-17:00 room S1*

**Analysseminariet****Grigori Rozenblioum**Pseudodifferential operators, C*-algebras and index formulas.

**Abstract:**The forth (and, I hope, the last) lecture in the cycle. The analytical machinery introduced in the previous lecture and some more algebra and topology are used to obtain index formulas for pseudodifferential operators with isolated singularities.*Tuesday Mar 9 at 15:15-17:00 room S1*

**Analysseminariet****Lars Andersson**Elliptic-hyperbolic systems and Einstein's equations.

**Abstract:**I will discuss some aspects of gauge fixing for the Einstein equations and some quasilinear elliptic-hyperbolic systems arising from the gauge fixed Einstein equations, with and without symmetry.*Tuesday Mar 2 at 15:15-17:00 room S1*

**Analysseminariet****Grigori Rozenblioum**Pseudodifferential operator, C*-algebras and index formulas (3).

**Abstract:**The third lecture in the cycle. We give some simplified formulations of the Atiyah-Singer index theorem. After this, we try to repeat the whole previous construction, but starting not with differential operators with smooth coefficients but with operators with discontinuous coefficients. Everything falls apart, but the Mellin transform saves the game miraculously. We find, as a result, the symbol algebra, which, unlike the continuous case, turns out to be non-commutative. An important role in the construction is played by Toeplitz operators with operator-valued symbols. Analysis of such operators involves generalized determinants. I hope to obtain the index formula for this case in the next (last) lecture of the cycle. As before, we are going to travel: analysis-algebra-functional analysis-harmonic analysis-algebra-topology.*Tuesday Feb 23 at 15:15-17:00 room S1*

**Analysseminariet****Lyudmila Turowska**Linear operator equations and related topics.

**Abstract:**The talk contains some generalizations of the well-known theorem of Fuglede and Putnam and Kleinike-Shirokov theorem on equivalence of the linear operator equations AX-XB=0 and A^{*}X-XB^{*}=0 and the equations [A,X]=0 and [A,[A,X]]=0 with normal coefficients A and B. We discuss linear equations of more general form on an algebra of operators and connection between these questions and theory of representations of *-algebras and harmonic analysis (problems of synthesis of ideals in Banach algebras and ordinary spectral synthesis in**R**^{n}).*Tuesday Feb 9 at 15:15-17:00 room S1*

**Analysseminariet****Grigori Rozenblioum**Pseudodifferential operator, C*-algebras and index formulas (2).

**Abstract:**Lecture 2 in the cycle. In the first lecture the basic geometrical information was given, enabling one to define the *-algebra of zero order operators and *-algebra of their symbols. In the second talk we discuss the relation of these notions to singular integral operators, define pseudodifferential operators and put these notions into C*-algebra contents. I hope to reach the formulation of the Atiyah-Singer index theorem and discuss important examples. After this, there will a 2-weeks break in the cycle.*Tuesday Feb 2 at 15:15-17:00 room S1*

**Analysseminariet****Grigori Rozenblioum**Pseudodifferential operator, C*-algebras and index formulas (1).

**Abstract:**In a cycle of appr. 4 talks I'll try to explain how natural considerations from functional analysis and algebra lead to the notation of pseudodifferential operators and the algebra of the symbols. The classical Atyiah-Singer index teorem will be explained in this context. After this I'll describe the generalization of this approach to operators having discontinuities in symbols.*Tuesday Jan 26 at 15:15-17:00 room S1*

**Analysseminariet****Magnus Wängefors**Rieszoperatorer på en Liegrupp av rang 2

**Abstract:**Genom att ta produkten av den affina gruppen med sig själv får man en Liegrupp av rang 2. Den har en naturlig Laplaceoperator, till vilken hör ett antal Rieszoperator. Vi koncentrerar oss på andra ordningens Rieszoperatorer och diskuterar deras L^{p}-begränsningar.*Tuesday Jan 19 at 15:15-17:00 room S1*

**Analysseminariet****Peter Sjögren**Rieszoperatorer på den affina gruppen

**Abstract:**Den affina gruppen är en tvådimensionell Liegrupp, som har en naturlig Laplaceoperator. Till den hör Rieszoperatorer, och vi studerar deras L^{p}-begränsning. Första ordningens Rieszoperatorer visar sig vara svårare att uppskatta än andra ordningens.*Tuesday Jan 12 at 15:15-17:00 room S1*

**Analysseminariet****Maria Roginskaya**: Characteristisation of existence of asymptotic lower estimates on singular measures for homogeneous Fourier multipliers on wk-H^{1}(Del 2).

**Abstract.**In the previous talk I presented an exact characteristisation of the existence of asymptotic lower estimates of the action of homogeneous Fourier multipliers on wk-H^{1}on finite measure via the singular parts of the measure and proved its sufficiency. Now the necessity will be proved in a constructive way.*Tuesday Dec 15 at 15:15-17:00 room S1*

**Analysseminariet****Maria Roginskaya**: Characteristisation of existence of asymptotic lower estimates on singular measures for homogeneous Fourier multipliers on wk-H^{1}.

In the talk (first of the two expected) we will present an exact characteristisation of the existence of asymptotic lower estimates of the action of homogeneous Fourier multipliers on wk-H^{1}on finite measure via the singular parts of the measure. The sufficiency will be proved even for a wider class of Fourier multipliers, which was mentioned in a previous talk.*Tuesday Dec 8 at 15:15-17:00 room MD10*

Wednesday Dec 9 at 15:15-17:00 room S2

**Analysseminariet och Algebraseminariet****Prof. L.L.Vaksman (Institute for Low Temperature, Kharkov)**:On function theory for some quantum boundary symmetric domains.

Abstract: We are going to discuss a quantum analogue of the classical calculi. We will describe integral representation of "q-functions" on the quantum unit disc (first lecture) and other Cartan domains (second lecture).*Tuesday Dec 1 at 15:15-17:00 room S1*

**Analysseminariet****Maurice de Gosson**:Lagrangian analysis and the Arnold-Leray-Maslov index.

Abstract: Semiclassical mechanics, whose ancestor is the physicists' WKB method, emerges when one tries to approximate the solutions to the time-dependent Schrödinger equation by neglecting some parameters (for instance Planck's constant, or the inverse of the mass). Semiclassical mechanics is actually harmonic analysis in phase space, more precisely, the study of the action of the metaplectic group on a certain type of half-forms defined on Lagrangian submanifolds of phase space. The rigorous definition of that action necessitates the use of a cohomological object, the ALM index, which has many useful applications in various other areas of mathematics, for instance in spectral flow theory. These considerations ultimately lead to the definition of a new mathematical structure, intimately related to Jean Leray's "Lagrangian Analysis".*Tuesday Nov 24 at 15:15-17:00 room S1*

**Analysseminariet****Johan Råde**: Asymptotics for elliptic uniformly degnerate operators (2)

Abstract: I will discuss the asympotic behaviour near the boundary of solutions to*Lu*=0 for elliptic uniformly degenerate operators*L*. I will discuss both the case of constant and the case of variable indicial roots.*Tuesday Nov 17 at 15:15-17:00 room S1*

**Analysseminariet****Johan Råde**: Asymptotics for elliptic uniformly degnerate operators

Abstract: I will discuss the asympotic behaviour near the boundary of solutions to*Lu*=0 for elliptic uniformly degenerate operators*L*. I will discuss both the case of constant and the case of variable indicial roots.*Tuesday Nov 10 at 15:15-17:00 room S1*

**Analysseminariet****Vilhelm Adolfsson**: Monotonicity formulas and unique continuation at the boundary

Abstract: We will show that the normal derivative of a harmonic function which vanishes on an open subset of the boundary of a Dini domain, can not vanish on a subset of positive surface measure.*Tuesday Oct 20 at 15:15-17:00 Room S1*

**Analysseminariet****Peter Sjögren**: Convergence for the square root of the Poisson kernel.

Abstract: In the ordinary Poisson integral in the unit disc, one replaces the kernel by its square root. This produces eigenfunctions of the hyperbolic Laplacian. After normalization they converge to the given boundary function. For an*L*^{p}boundary function, this convergence takes place in an approach region which is wider than any nontangential cone, and which gets wider with increasing*p*(results by Sjögren and Rönning). We shall see which approach regions occur for .*Tuesday Oct 13 at 15:15-17:00 Room S1*

**Analysseminariet**

**Nikolaos Bournaveas**: Low regularity local solutions of the Dirac Klein-Gordon equations.

Abstract: The classical local existence theorem for nonlinear wave equations of the form requires that the initial data belong to with . The lower bound for*s*can be significantly improved by using the space time estimates of Strichartz and Brenner. If the nonlinear terms have a special structure, as is the case for most of the equations that arise in Mathematical Physics, this lower bound can be impoved even further by using recent estimates of Klainerman and Machedon. We shall discuss all these estimates as well as applications to the Dirac Klein-Gordon equations.*Thursday Oct 08 at 15:15 - 17:00 Room S3*

**Analysseminariet:**, Bilbao:

Luis Vega

Some remarks on illposedness for the Geometric KdV

**Abstract:**We will show that there is no uniform continuity for the solution of the initial value problemu

_{t}+u_{xxx}+ ¦u¦^{2}u_{x}= 0u(x,0) = u

_{0}(x)if the initial datum is not regular enough.

*Tuesday Oct 06 at 15:15 - 17:00 Room S1*

**Analysseminariet:**:

Maria Roginskaya

Multi-dimensional Riesz sets and some problems of engineering.

**Abstract:**The classical theorem of F. and M. Riezs says (in particular) that if a finite measure has a Fourier transform which vanishes on the negative numbers, then this measure has no singular part. Some generalizations of this statement to both one- and multi-dimensional cases will be considered in the first part of the talk. In the second part I will talk (in quite a naive way) on an antenna construction and state a mathematical problem which arises naturally.*Tuesday Sept 29 at 15:15 - 17:00 Room S1*

**Analysseminariet:**, University of California at Davis:

Naoki Saito

The Least Statistically-Dependent Basis and Its Applications.

**Abstract:**Statistical independence is one of the most desirable properties for a coordinate system for representing and modeling signals and images. In reality, however, truly independent coordinates may not exist for a given set of images, or it may be computationally too difficult to obtain such coordinates. Therefore, it makes sense to obtain the least statistically dependent coordinate system efficiently. This basis---we call it "Least Statistically-Dependent Basis" (LSDB)---can be rapidly computed by minimizing the sum of the differential entropy of each coordinate in the basis library. This criterion is quite different from the Joint Best Basis (JBB) proposed by Wickerhauser. We demonstrate the use of the LSDB for signal and image modeling and compare its performance with JBB and Karhunen-Loeve Basis (KLB).*Tuesday Sept 22 at 15:15 - 17:00 Room S1*

**Analysseminariet:**, St. Petersburg:

Evguenia Malinnikova

Some properties of harmonic differential forms.

**Abstract:**We consider harmonic differential forms as a generalization of complex analytic funtions. We will discuss approximation theorems for harmonic differential forms on Riemannian manifolds. Also the three-spheres theorem for harmonic forms will be proved.*Monday Sept 21 at 15:30 - 17:00 Room Hörsalen*

**Analysseminariet:**, Rom:

Fausto Di Biase

P. Fatou meets H. von Koch.

**Abstract:**McMillan's twist theorem (1969) states that every conformal map of the unit disc into the complex plane is either conformal or twisting at almost all points of the boundary. In this preliminary report, we describe some results obtained in an attempt to interpret this result from the point of view of the theory of the boundary behaviour of harmonic functions.*Tuesday Sept 8 at 15:25 - 17:00 Room S1*

**Analysseminariet:**:

Grigori Rozenblioum

Domination, majoration and eigenvalue estimates.

**Abstract:**Let*K*and*L*be two integral operators with kernels, respectively,*K*(*x*,*y*) and*L*(*x*,*y*). We say that*K*majorates*L*if almost everywhere. The old question is which properties of the operator*K*in*L*-space (or, more generally in_{2}*L*_{p}) are inherited by*L*. It is obvious that boundedness is inherited, somewhat more complicated is that compactness is inherited as well. In*L*, it is easy to show that the property of the operator to belong to the ideal_{2}*B*_{p}(consisting of operators*T*for which the sequence of eigenvalues of*T*belongs to^{*}T*l*_{p/2}) is inherited, provided*p*is an even integer; this takes place, in particular, for the Hilbert-Schmidt class*B*. However this becomes wrong if_{2}*p*is not an even integer - and the corresponding construction by V.Peller (1980) involves quite striking results from the harmonic analysis. After this, the general opinion about inheritance of eigenvalue estimates by majorated operators was rather pessimistic. In the talk we will discuss the above results and show that quite a number of spectral estimates are inherited, provided the operators*K*and*L*are related, respectively, to a positivity preserving semi-group and to a dominated semi-group. From this general fact, it follows, in particular, that virtually all eigenvalue estimates for the Schrödinger operator are carried over automatically to the magnetic Schrödinger operator, regardless of the method used in obtaing these estimates.*Tuesday May 26 at 15:15 - 17:00 Room S1*

**Analysseminariet:**:

Peter Sjögren

Maximalfunktionen för Ornstein-Uhlenbeck-halvgruppen i ändlig och oändlig dimension.

**Abstract:**I**R**^{d}ersätter man Lebesguemåttet med Gaussmåttet. Motsvarigheten till värmeledningshalvgruppen blir då den s.k. Ornstein-Uhlenbeck-halvgruppen.*Tuesday May 19 at 15:15 - 17:00 Room S1*

**Analysseminariet:**Politecnico di Milano:

Stefano Meda,

On the Kunze--Stein phenomenon

**Abstract:**(Click here)*Tuesday May 12 at 15:15 - 17:00 Room S1*

**Analysseminariet:**

Johan Råde:

Likformigt degenererade operatorer 4

**Abstract:**Fortsättning på seminarierna från mars och april.*Tuesday May 05 at 13:15 - 15:00 Room S1*

**Analysseminariet:**Rom:

Fausto Di Biase,

Boundary behaviour of bounded harmonic functions in the unit disc along tangential curves

**Abstract:**We study the boundary behaviour of bounded harmonic functions in the unit disc along curves tangential to the boundary; the shape of the curves may change from point to point. The results are joint work with A. Stokolos, O. Svensson and T. Weiss.*Tuesday April 28: at 15:15 - 17:00 Room S1*

**Analysseminariet:**

Nils Dencker, Lund:

Lokal lösbarhet av differentialoperatorer

**Abstract:**På 50-talet så var den allmänna uppfattningen att alla differentialoperatorer*P*är lokalt lösbara, dvs att ekvationen*Pu = f*har en lokal lösning för alla i ett underrum av ändlig kodimension. Detta hade också bevisats vara sant för partiella differentialoperatorer med konstanta koefficienter och för första ordningens partiella differentialoperatorer med variabla reella koefficienter. Det var därför en sensation då Hans Lewy 1957 presenterade ett exempel på en naturlig första ordningens partiell differentialoperator med variabla komplexa koefficienter i

**R**^{3}, som inte är lokalt lösbar någonstans.

Detta ledde till en snabb utveckling av teorin under de följande 20 åren. De senaste åren har nya framsteg gjorts, men huvudproblemet har förblivit olöst: att finna nödvändiga och tillräckliga villkor för att en operator ska vara lokalt lösbar.

Föredraget avser att ge en historisk presentation av lösbarhetsproblemet, presentera de framsteg som har gjorts inklusive några av föreläsarens egna resultat inom området. Föredraget avser att vara relativt elementärt och inte förutsätta mer än grundläggande förkunskaper i distributionsteori och Fourieranalys.*Tuesday April 14: at 15:15 - 17:00 Room S1*

**Analysseminariet och KASS:**

John Wermer, Brown University:

Linking numbers and boundaries of analytic varieties.

*Tuesday April 7: at 15:15 - 17:00 Room S1*

**Analysseminariet:**

Johan Råde

Uniformly degenerate operators, a new approach 3.

**Abstract:**This is a continuation of the talk I gave March 31.*Tuesday March 31: at 15:15 - 17:00 Room S1*

**Analysseminariet:**

Johan Råde

Uniformly degenerate operators, a new approach 2.

**Abstract:**This is a continuation of the talk I gave March 13. I will finish the representation theoretic aspects of the problem. Then I will discuss some background material on Fredholm properties of ordinary differential operators acting on weighted Sobolev spaces on the real line. These Lemmas will be used in the third and last part of the talk, where I establish regularity and existence for elliptic uniformly degenerate operators.*Tuesday March 17 at 15:15 - 17:00 Room S1*

**Analysseminariet:**:

Bengt Alrud

Bernstein-type functions

**Abstract:**In this talk I will characterize some functions which are not necessarily negative definite functions, but admit a Lévy-Khintchine formula. These functions include the completely alternating ones and thereby the Bernstein functions.*Friday March 13: at 13:15 - 15:00 Room S1*

**Analysseminariet:**

Johan Råde

Uniformly degenerate operators, a new approach

**Abstract:**I will describe a new proof of regularity and existence for elliptic uniformly degenerate partial differential operators on domains with boundary. The method is similar to the classical method for proving regularity and existence for elliptic partial differential operators, except that we use the Fourier transform on a noncommutative group instead of the usual Fourier transform on**R**^{n}*Tuesday March 10 at 15:15 - 17:00 Room S1*

**Analysseminariet:**, Uppsala universitet:

Anders Öberg

Convergence of the transfer operator for iterated function systems

**Abstract:**Usually iterated function systems are equipped with weights which in some sense determine the role each map play. If, in particular, the weights sum to one at every point of the state space, then we may interpretate each weight as the probability with which we choose the corresponding map. If we deal with conformal maps it is natural to consider other, non-probabilistic, weights. We will prove some theorems concerning uniqueness and approximation of invariant measures when the weights (weight functions) are rather arbitrary. We will also discuss the related problem of approximating infinite dimensional Perron-Frobenius theory with finite dimensional Perron-Frobenius theory.

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