15/07/2004, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Reinhard Farwig, Darmstadt University of Technology, Germany
-Analysis of Viscous Fluid Flow Past or Around a Rotating Obstacle
Consider the problem of time-periodic solutions of the Stokes and Navier-Stokes system modelling viscous incompressible fluid flow past or around a rotating three-dimensional obstacle. Introducing a rotating coordinate system attached to the body a linearization yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. This system admits a more or less explicit solution for the whole space problem when the axis of rotation is parallel to the velocity of the fluid at infinity. For the analysis of this solution in
-spaces we will use tools from harmonic analysis and a special maximal operator reflecting paths of fluid particles past or around the obstacle.
25/06/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Evgeny Semenov, Voronezh State University, Russia
The Banach-Saks Index
We introduce and study the concept of the Banach-Saks index. It is motivated by the Banach-Saks and -Banach-Saks property. The Banach-Saks index is related with main geometrical characteristics of Banach functional spaces ( -convexity and -concavity, Rademacher type, Boyd indices). We study operators related with the Banach-Saks index.
21/05/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro
Weighted Sobolev Theorems for Spatial and Spherical Potentials
inthe Lebesgue Spaces with Variable Exponent
One of the open problems in the "variable exponent business" was
related to embeddings in the Sobolev spaces with variable exponent
in the case of unbounded domains, in particular, in the case of the
whole Euclidean space. As is known, such embeddings are related to
mapping properties of potential type operators. In this talk there
are presented weighted results on the boundedness of the Riesz
potential operator from the generalized Lebesgue space over
Euclidean space, with variable exponent \(p(x)\), to a similar
space with the Sobolev limiting exponent \(q(x)\).
Spherical potential operators are also treated in a similar
setting in the corresponding spaces with variable exponent on the
unit sphere in the Euclidean space. Stereographical projection is
used for this purpose, which maps the Euclidean space
\(\mathbb{R}^n\) onto the unit sphere \(S^n\) in
\(\mathbb{R}^{n+1}\). One of the remarkable properties of this
mapping is that it transforms the distance between two points \(x\)
and \(y\) in \(\mathbb{R}^n\) exactly into the difference between
their images \(s(x)\) and \(s(y)\) on \(S^n\) multiplied by the
power weight functions fixed to infinity. This property allows to
derive many results for various types of operators, known for
\(\mathbb{R}^n\) to similar types of spherical operators on the
sphere, and, to the contrary, from what may be obtained on the
compact set \(S^n\), one may derive results for operators on
\(\mathbb{R}^n\), which is a non-compact set (with respect to the
usual metrics). The talk is based upon joint work with Boris
Vakulov.
14/05/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Ana Moura Santos, Instituto Superior Técnico, U.T. Lisboa
Regularity of Solutions to a Diffraction Problem with Oblique
Derivatives on a Strip
We consider a boundary-transmission problem for the Helmholtz
equation, in a Bessel potential space setting, which arises within
the context of wave diffraction theory. The boundary under
consideration consists of a strip, and certain conditions are
assumed on it in the form of oblique derivatives. Those are of
particular importance from the physical point of view in the
context of materials involving non-homogeneous impedances in the
boundary. The well-posedness of the problem is shown for a range of
non-critical regularity orders of the Bessel potential spaces,
which include the finite energy norm space. In addition, an
operator normalization method is applied to the critical orders
case.
02/04/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Nikolai Nikolskii, Université de Bordeaux, France, and Steklov Institute of Mathematics, St. Petersburg, Russia
Some Unsolved Problems around
and Operator Theory
We give an account and discuss some open problems related to the following topics:
- Efficient inversions (for the Wiener algebra, Wiener-Hopf operators, and quotient algebras of
);
- Weak generators in quotient algebras of
;
- Operator corona problem and approximation property.
26/03/2004, 15:15 — 16:15 — Sala P3.10, Pavilhão de Matemática
Cláudio Fernandes, Universidade Nova de Lisboa
Símbolos para Classes de Álgebras de Operadores de
Tipo Não Local
Using the local-trajectory method we construct symbol calculi for
extensions of an algebra of bounded linear operators, generated by
operators of multiplication by slowly oscillating piecewise
continuous functions and convolution type operators, by unitary
operators associated with shifts with respect to contour and dual
(in Fourier sense) variables.
26/03/2004, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Torsten Ehrhardt, Technical University of Chemnitz, Germany
Factorization Theory for Singular Integral Operators with Flip
The invertibility of singular integral operators with flip on the
unit circle is equivalent to the invertibility of
Toeplitz-plus-Hankel operators with a matrix symbol of a particular
structure. We show how to develop a factorization theory for such
Toeplitz-plus-Hankel operators. The kind of factorization differs
from the usual generalized Wiener-Hopf factorization, but is
related to it. Some examples where the factorization can be
constructed explicitly will be also discussed.
12/03/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Maria do Carmo Martins, Instituto Superior Técnico, U.T. Lisboa
Factorization of Triangular Symbols - The Trinomial Case
We study the solutions of Riemann-Hilbert problems with certain
triangular matrix symbols in appropriate spaces of analytic
functions. Assuming that a solution to this Riemann-Hilbert problem
is known, we show that, in certain cases, it is possible to
determine a second solution to the same problem and to check
whether these two solutions yield the factors of a canonical
factorization of the symbol using only very simple ideas.
05/03/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Michael A. Semenov-Tian-Shansky, Université de Bourgogne, Dijon, France
Q-Deformed Quantum Toda Lattice: Modular Duality, Separation of
Variables, and Baxter Equations
Quantum Toda lattice provides a link between representation theory
of semisimple Lie groups and quantum inverse scattering method; its
q-deformed version is particularly interesting, since it gives a
guide to the representation theory of non-compact quantum groups.
This theory is still not fully understood; the study of the
q-deformed Toda lattice reveals interesting new phenomena in
representation theory : modular duality, first discovered by
Faddeev a few years ago (the emergence of a "second" quantum group
which is acting in the same representation space and centralizes
the action of the first one), and points to the importance of a
special class of meromorphic functions (Barnes double sine
functions and their relatives). The spectral problem for the
q-deformed Toda lattice is inductively reduced to a system of
finite difference functional equations (Baxter equations).
13/02/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Instituto Superior Técnico, U.T. Lisboa
Commutatorsof Singular Integrals on Variable \(L_p\) Spaces II
This talk is a continuation of the previous one by Andrei
Lerner. We will show that if a function \(b\) belongs to the
Zygmund space \(L\log L\) locally and the commutator \([b,T]\) with
the Calderon-Zygmund operator \(T\) is bounded on the variable
\(L_p\) space, then \(b\) is of bounded mean oscillation. This is a
necessry part of our generalization of the Coifman-Rochberg-Weiss
commutator theorem. Certainly, the variable exponent p in
our theorem has to satisfy some (natural) assumptions. This talk is
based on the joint work with Andrei Lerner (Bar-Ilan University,
Israel).
06/02/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro
Hardy-Littlewood-Stein-Weiss Inequality in the Lebesgue Spaces with
Variable Exponent
The Hardy-Littlewood inequality, for one-dimensional fractional
integrals for Lebesgue spaces in the case of power weights and the
limiting exponent, was generalized to potential type operators by
Stein and Weiss. In the rapidly developing "variable exponent
business" there was an open problem to prove such an inequality for
potentials of variable order in the weighted Lebesgue spaces with
variable \(p(x)\), that is to prove the boundedness of the
potential type operator of order \(m\) from the weighted Lebesgue
space of order \(p(x)\) to a weighted space of order \(q(x)\) with
\(1/q(x) = 1/p(x)-m/n\).
The solution of this problem is presented for the case of
bounded domains in the Euclidean space. It is based on a technique
of estimation of weighted norms in the Lebesgue spaces with
variable exponent of powers of distances \(|y-x|\) truncated to the
exterior of the ball of radius r centered at the point
x of the Euclidean space, and on Hedberg's approach of
comparison of potentials with maximal functions.
One of the main points in the result obtained is that the bounds
for the weight exponents are exactly related to the values of the
Lebesgue exponent \(p(x)\) at the points to which the weight is
fixed. As a corollary, imbeddings for the Sobolev spaces with
varying \(p(x)\) are derived.
30/01/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Andrei K. Lerner, Bar-Ilan University, Israel
Commutatorsof Singular Integrals on Variable \(L_p\) Spaces I
This talk is a continuation of the previous one by Andrei
Lerner. We will show that if a function \(b\) belongs to the
Zygmund space \(L\log L\) locally and the commutator \([b,T]\) with
the Calderon-Zygmund operator \(T\) is bounded on the variable
\(L_p\) space, then \(b\) is of bounded mean oscillation. This is a
necessry part of our generalization of the Coifman-Rochberg-Weiss
commutator theorem. Certainly, the variable exponent p in
our theorem has to satisfy some (natural) assumptions. The talk is
based on the joint work with Alexei Karlovich.
19/12/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Yuri I. Karlovich, Universidad Autónoma del Estado de Morelos, México
C*-algebras of Integral Operators with Shifts having Massive Sets
of Periodic Points
A general scheme to study C*-algebras of bounded linear operators
with discrete groups of unitary shift operators is considered. This
scheme is based on applications of spectral projections invariant
under action of the group of shifts. Several applications of this
scheme to concrete C*-algebras of integral operators with shifts
are obtained.
21/11/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Helena Maria Narciso Mascarenhas, Instituto Superior Técnico, U.T. Lisboa
Operadores de Convolução em Cones e Álgebras Standard Model
Consideram-se operadores de convolução em cones com símbolo
na álgebra de Wiener. É uma questão em aberto saber se nesta
classe, a propriedade de Fredholm garante a invertibilidade dos
operadores. Propõe-se um método para determinar a dimensão do
núcleo deste tipo de operadores quando são de Fredholm. Com este
propósito, estuda-se uma álgebra que contém sucessões de
aproximação do operador e que é uma álgebra standard model, o
que permite obter uma relação entre os valores singulares dessa
sucessão de aproximaçãso e a dimensão do núcleo do
operador.
14/11/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Pedro A. Santos, Instituto Superior Técnico, U.T. Lisboa
On Asymptotics of Toeplitz Determinants
Szegö's strong limit theorem for Toeplitz determinants is
connected to models in statistical physics for the behaviour of
ferromagnets. We obtained a proof for Szegö's strong limit
theorem for Toeplitz operators with a symbol having a nonstandard
smoothness. It is assumed that the symbol belongs to the Wiener
algebra and, moreover, the sequences of Fourier coefficients of the
symbol with negative and nonnegative indices belong to weighted
Orlicz classes generated by complementary N-functions both
satisfying the delta2 condition.
07/11/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Frank-Olme Speck, Instituto Superior Técnico, U.T. Lisboa
Diffraction by Rectangular Wedges with Different Face Impedances
Some basic hitherto open boundary value problems are treated by
a new potential approach, using so-called reproducing half-line
potentials. It involves symmetry properties, operator relations,
factorization methods and normalization techniques for convolution
type operators with symmetry that were recently developed by
various members of the CEMAT operator theory group. Interior wedge
problems of normal type for the Helmholtz equation could be
completely analyzed and explictly solved by analytical formulas
which also allow fine results about the solutions' regularity
properties. It turns out that these results are also good for
reducing exterior wedge problems in a most efficient way to scalar
boundary pseudodifferential equations. However, their nature is
much more complicated and cumbersome compared with the class of
interior wedge diffration problems. Only very particular boundary
conditions allow a detailed comparable analysis. The lecture is
based upon common work with L. P. Castro and F. S. Teixeira.
16/10/2003, 15:00 — 16:00 — Sala P3.31, Pavilhão de Matemática
Yu. I. Karlovich, Universidad Autónoma del Estado de Morelos, México
Algebras of Singular Integral Operators with Shifts and Oscillating
Coefficients
The talk is devoted to the Fredholm theory of algebras of singular
integral operators with coefficients admitting piecewise slowly
oscillating discontinuities and with discrete subexponential groups
of piecewise smooth shifts acting topologically free on Lebesgue
spaces over composed contours. A general local method of studying
the Fredholmness of nonlocal bounded linear operators on Banach
spaces and the limit operators techniques are applied.
10/10/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Maria Amélia Bastos, Instituto Superior Técnico, U.T. Lisboa
Symbol Calculus for Banach Algebras with Slowly Oscillating and
Piecewise Continuous Data
A symbol calculus is constructed for two different algebras of
bounded linear operators generated by operators of multiplication
by slowly oscillating piecewise continuous functions and
convolution type operators. The methods for obtaining the symbol
are compared for the settings of Banach algebras and C* algebras.
03/10/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Ana Moura Santos, Instituto Superior Técnico, U.T. Lisboa
Wave Diffraction by a Strip Grating: The two-straight line approach
The present work deals with the problem of wave diffraction by a
periodic strip grating which occurs in several application areas,
in particular, antenna and waveguide problems, in electrical
engineering, and diffraction of sound waves by periodic screens, in
acoustics. We give a rigorous formulation of boundary-value
problems of wave diffraction by a periodic strip grating, in which
the width of each strip can be different from the spacing between
any two adjacent strips, and also, greatly simplify the study of
the invertibility of the operator which is associated with the
diffraction problem restricted to Dirichlet and Neumann boundary
conditions when the period is equal to double the width of the
strips. The equivalence of the operators that appear in the
original formulation of the diffraction problems to a Toeplitz
operator defined on a space of functions with a two-straight line
domain allows us to give sufficiently simple formulas for the
inverse of the operator when the period of the grating is equal to
double the width of the strips. This is a joint work with Prof.
Amélia Bastos e Prof. Ferreira dos Santos.
26/09/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Luís Filipe Pinheiro de Castro, Universidade de Aveiro
Invertibilidade de Operadores de Convolução Compostos com
Operadores de Extensão Par ou Ímpar
São analisados, sob a perspectiva da invertibilidade,
operadores que centralmente se descrevem pela composição de
operadores de convolução (na recta real) com operadores de
extensão par ou ímpar (da semi-recta positiva para a recta real).
Tais operadores são definidos entre produtos de espaços de
potenciais de Bessel e possuem pré-símbolos numa classe
dependente das funções matriciais contínuas no sentido de
Hölder. Diferentes formas de factorizar estas funções matriciais
levam ao estabelecimento de condições que permitem
representações dos inversos dos operadores em estudo.