- 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 S1
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. We show how operators with an attractive
delta-potential supported by a graph can be modeled in the strong
resolvent sense by point-interaction Hamiltonians. The result
is illustrated on finding the spectral properties for two simple
examples with the graph being a circle and a star, respectively.
Furthermore, we use this method to search for resonances due
to quantum tunneling or repeated reflections.
- 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 Vi diskuterar asymptotiska egenskaper
hos rum-tidmedelvärden, (i lämpliga Sobolev- och Besovrum),
av lösningar till den ickelinjära Klein-Gordonekvationen
och under vilka förutsättningar dessa egenskaper "ärvs"
från lösningarna av motsvarande, (samma data), linjära
Klein-Gordonekvation. Resultat av detta slag är viktiga
verktyg, bl. a. då man visar existensen av överallt
definierade spridningsoperatorer och entydigheten hos svaga lösningar.
Vi diskuterar även graden av avtagande för rum-tidmedelvärden
av lösningar till den icke-linjära Klein-Gordonekvationen.
Lösningarnas avtagandeegenskaper kan kopplas till avtagandeegenskaper
och graden av avtagande för energi och lokal energi.
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- Tuesday 21 October at 15.15, room S1
Vladimir A. Mikhailets, Kiev: Common Eigenvalue problem
and periodic Schrödinger operator
Abstract
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- 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.
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- Tuesday 23 September at 15.15, room S1
Maria Roginskaya: Energy dimension via Fourier transform:
Applications and Generalizations (joint with K. E. Hare)
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.
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- 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
Markus Kunze, Essen: Existence of the best constant for the
Strichartz inequality.
Abstract
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- Tuesday 26 August at 15.15, room S1
Joachim Hilgert: On the Lewis functional equation
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.
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- 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.
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- 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
Hiroaki Aikawa, Shimane University, Japan: Fatou and Littlewood
theorems for Poisson integrals with respect to non-integrable
kernels
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.
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- 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.
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- 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.
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- 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.
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- 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.
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- Tuesday 4 March at 15.15, room S1
Florian Vasilescu, Lille: Moment Problems on Semi-Algebraic
Sets and Applications
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.
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- 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.
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- 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.
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- 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
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Seminars in 2002
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- 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.
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- 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
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- 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.
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- Thursday, October 17 at 15.15 Hörsalen
Aline Bonami, Orleans, som besöker oss för
att promoveras till hedersdoktor vid GU: An application
of Fourier analysis: modelisation of X-Ray images of
bones and directional asymptotic self-similarity
Tuesday, October 15 at 15.15 room S1
Giancarlo Mauceri, Genova: Holomorphy of spectral
multipliers of Lp
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.
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- 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, Rd=RmxRn,
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 L1-functional
calculus for self-adjoint operators
Abstract.
- 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).
Abstract.
- 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
Abstract:
- 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={x1,...,xn}
of points with intensities ² =(²1,..., ²n)
is defined by the differential expression -d^2/dx^2 on
functions Æ (x) that belong to the Sobolev space W22(R\
X) and satisfy, in points of the set X, the following
conjugation conditions:
Æ'(xk+0)= Æ'(xk-0), Æ(xk+0)
- Æ(xk-0) = ²k Æ'(xk).
Let all the intensities of ´'-interactions satisfy ²k
<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
R1, of Lebesgue measure zero, with a Borel
measure.
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 Lp
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 L2 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 L2 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±(H,-¿)
exist and are asymptotically complete.
- 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 Ln to L.
In the sixties Pucci suggested the precise Lp-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 cnfn, where cn2
< and {fn} is an orthogonal system in
L2[0,1]. Such a series converges in L2,
but we are interesting in convergence or summability a.e. on
[0,1].
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 Ht of
positive order 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: L2(X,m)
L2(Y,n) with the kernel 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
L2, it is always possible to consider the so-called
Riesz means of A, which are special spectral multipliers
involving a real parameter. One important question is whether
the Riesz means are uniformly bounded operators on Lp.
In our case the A's will be certain second order differential
operators on R2n, 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.
- 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 Lp 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 Lp 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 Lp replaced by a class of Orlicz
spaces that resemble Lp. 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 Rd
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 Cn. 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
He 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)-1dy/dx+Q(x)y) + Q*(x)dy/dx+R(x)y=»H(x)y,
(2)
where A, Q, R, H Lloc( R) and A(x)
is positive definite for all x R and 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,|yn- xn|}, which is a natural infinite
dimensional extension of the n-dimensional Euclidian spaces.
In this setting we consider product sets \prod En,
where the coordinate set En 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 En)=\sum dim(En),
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,
L1(G) be the group algebra, and A(G)
be the Fourier algebra of G. If G is Abelian,
A(G)<-> L1(G^) via the Fourier transform,
where 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 L1(G^), even though we may not have
a reasonable candidate for ``G^''.
B. Johnson showed that L1(G) is amenable
(i.e. every derivation of L1(G) into a dual
Banach L1(G)-module is inner) precisely when
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-1/2
and L-1/2X, where X is an invariant
vector field. The question is whether they are bounded on Lp.
The main difficulty is the global behaviour of these operators.
- 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 Rn
R which for some T > 0 belong to Hs(Rn)
on (0,T) and the space of functions which belong to
H1 on a fixed interval (0,T0).
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.
- 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 C6 and H5-Smooth Almost Spherical Domains
Abstract:It is a joint work with professor N.Shirokov.
Let D c Cn be an almost spherical domain with
C6-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 BD(z,
z) as 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 C6
to H5 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 L1 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 Lp, 1<p d
,
and weak Lp, 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)-1
(a'a-1)dx
for the winding number of the curve in the complex plane described
by a function 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 KapustinReflexivity 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 Ha 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 Ha 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 an=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, Cn,
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 L2 onto H2) maps Lp onto
Hp for 1, has weak type (1,1), and is discontinuous on
L1 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 Rn.
Abstract: We wish to show that the complex structure for
Rn 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*(Hp)
of the Hardy space Hp. We investigate measures
on the unit disc D such that O*(Hp)cLp(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 L2-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 Lp(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
L2 for some measure space. If m is a bounded
Borel function on R, one can define a bounded operator
m(A) on L2 by means of the spectral resolution
of A. For 1Ø p<
, the
function m is called an Lp-multiplier
if m(A) extends to a bounded operator on Lp.
These multipliers form a Banach algebra. Necessary or sufficient
conditions for Lp-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 Lp 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, H1 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 H1 suitable for non doubling
measures. We will show how several classical results involving
BMO and H1 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 ball
Abstract: The real hyperbolic ball is the euclidean ball Bn
endowed with the hyperbolic geometry. Let Sn-1
be the unit sphere in Rn. Call D
the associated Laplacian and say that a function u on
Bn 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 L2-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
Rn).
- Tuesday Feb 9 at 15:15-17:00 room S1
Analysseminariet
- Grigori RozenblioumPseudodifferential 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 RozenblioumPseudodifferential 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 Lp-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 Lp-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-H1 (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-H1 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-H1.
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 Lp 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:
Luis Vega, Bilbao:
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 problem
ut+uxxx+ ¦uپux
= 0
u(x,0) = u0(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:
Naoki Saito, University of California at Davis:
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:
Evguenia Malinnikova, St. Petersburg:
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:
Fausto Di Biase, Rom:
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 L2
-space (or, more generally in L p) are
inherited by L. It is obvious that boundedness is inherited,
somewhat more complicated is that compactness is inherited as
well. In L2, it is easy to show that the property
of the operator to belong to the ideal Bp
(consisting of operators T for which the sequence of eigenvalues
of T*T belongs to lp/2
) is inherited, provided p is an even integer; this takes
place, in particular, for the Hilbert-Schmidt class B2.
However this becomes wrong if 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 Rd 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:
Stefano Meda, Politecnico di Milano:
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:
Fausto Di Biase, Rom:
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 R3, 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 Rn
- Tuesday March 10 at 15:15 - 17:00 Room S1
- Analysseminariet:
Anders Öberg, Uppsala universitet:
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.