25/07/2003, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Yu. I. Karlovich, Universidad Autónoma del Estado de Morelos, México
An Algebra of Pseudodifferential Operators with Symbols of
Restricted Smoothness
The talk is devoted to an algebra of pseudodifferential
operators with presymbols which slowly oscillate with respect to
the contour variable and are continuous functions of bounded total
variation with respect to the dual variable. Boundedness and
compactness conditions for such operators are obtained. A symbol
calculus is constructed on the basis of an appropriate
approximation of the presymbols by infinitely differentiable
functions and by use of the techniques of oscillatory integrals. An
isomorphism between the Banach algebra of pseudodifferential
operators and their symbols is established. A Fredholm theory for
the pseudodifferential operators under consideration is
developed.
18/07/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Cristina Câmara, Instituto Superior Técnico
Factorization of some Classes of Almost-Periodic Symbols
27/06/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
António Caetano, Universidade de Aveiro
Capacidade de Crescimento Local para Funções de Espaços do Tipo Besov e Triebel-Lizorkin
Usando o conceito de envelope de crescimento — recentemente introduzido por Haroske e Triebel — e após revisão de resultados previamente conhecidos e o seu possível interesse, mede-se a capacidade que as funções dos espaços de Besov e de Triebel-Lizorkin (com parámetros \(s, p, q\) no espaço euclideano de dimensão \(n\)) de derivação generalizada (tipo \(\Psi\)) têm para crescer localmente, tomando especial atenção ao caso crítico \(s = n/p\). Mostra-se, em particular, como a consideração deste caso permitiu unificar o tratamento dos casos subcrítico (\(s \lt n/p\)) e crítico, tanto no contexto clássico (\(\Psi\) idêntico a \(1\)) como no contexto generalizado. Neste último, naturalmente a função \(\Psi\) influencia os resultados a apresentar.
07/06/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
António Caetano, Universidade de Aveiro
Capacidade de crescimento local para funções de espaços do tipo Besov e Triebel-Lizorkin
Usando o conceito de envelope de crescimento — recentemente introduzido por Haroske e Triebel — e após revisão de resultados previamente conhecidos e o seu possível interesse, mede-se a capacidade que as funções dos espaços de Besov e de Triebel-Lizorkin (com parámetros \(s, p, q\) no espaço euclideano de dimensão \(n\)) de derivação generalizada (tipo \(\Psi\)) têm para crescer localmente, tomando especial atenção ao caso crítico \(s = n/p\). Mostra-se, em particular, como a consideração deste caso permitiu unificar o tratamento dos casos subcrítico (\(s \lt n/p\)) e crítico, tanto no contexto clássico (\(\Psi\) idêntico a \(1\)) como no contexto generalizado. Neste último, naturalmente a função \(\Psi\) influencia os resultados a apresentar.
06/06/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Ana Moura Santos, Instituto Superior Técnico
Minimal Normalization of Wiener-Hopf Operators and Applications to
Boundary Value Problems with Plane Discontinuities
A class of operators is studied which results from certain boundary
and transmission problems in the half plane and in the two-part
plane. For different orders of the boundary operators due to the
upper and lower banks these are often not normally solvable
problems. A classification of not normally solvable problems is
given for both geometrical situations and we apply the method of
minimal normalization in Bessel potential spaces in order to solve
some of the boundary value problems. The talk is mainly inspired by
a suggestion made by Prof. E. Meister and reflects some recent
results of a joint work with N. Bernardino.
16/05/2003, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Luís Pessoa, Instituto Superior Técnico, U.T. Lisboa
Álgebra-C* gerada por um número finito de projecções poly e
anti-poly Bergman com coeficientes seccionalmente contínuos
São estabelecidos critérios de Fredholm para operadores da
álgebra-C* gerada por projecções em espaços poly e anti-poly
Bergman com coeficientes seccionalmente contínuos. Não obstante
as generalizações dos espaços de Bergman introduzidas manterem o
contexto dos espaços de Hilbert de funções analíticas, são
necessárias proposições específicas relacionadas com
aproximação polinomial, facilmente obtidas no caso do disco
unitário.
O problema desenvolve-se em \(L_2\), o que, através de
localização, permite estabelecer propriedades genéricas
relacionadas com a estrutura algébrica e topológica da álgebra
considerada. Todas as álgebras locais são álgebras matriciais e
critérios de invertibilidade são obtidos por intermédio de um
resultado caracterizante de álgebras-C* particulares geradas por
projecções ortogonais e generalizando um resultado apresentado
anteriormente.
02/05/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Instituto Superior Técnico, U.T. Lisboa
Algebras of Functions with Fourier Coefficientsin Weighted Orlicz
Sequence Spaces
We prove that the set of all integrable functions whose sequences
of negative (resp. nonnegative) Fourier coefficients belong to a
two-weighted Orlicz sequence space forms an algebra under pointwise
multiplication. We show that this is a Banach algebra with the
factorization property.
11/04/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Yuri I. Karlovich, Universidad Autónoma del Estado de Morelos, México
Index Calculations for Convolution Type Operators
The talk deals with index formulas for convolution type operators
on Lebesgue spaces over the real line. In general, such operators
belong to the Banach algebras generated by the operators of
multiplication by piecewise continuous and oscillating matrix
functions and by the convolution operators with piecewise
continuous and oscillating matrix symbols being Fourier
multipliers.
04/04/2003, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Frank-Olme Speck, Instituto Superior Técnico, U.T. Lisboa
On a Class of Wedge Diffraction Problems Posted by Erhard Meister
A class of canonical wedge diffraction problems was formulated
by E. Meister in 1986 and subsequently treated by an operator
theoretical approach in various publications of his research group.
Certain subclasses of those problems, recognized of being unsolved,
are subject of the present talk. Some of them are now successfully
attacked by the help of
- operator relations,
- a new factorization approach for convolution type operators
with symmetry, and
- the method of minimal normalization in Sobolev spaces.
Several gaps of the existing theory are filled, new problems are
recognized reflecting the challenges of the present state of the
art. The talk is mainly based upon joint recent work with L. Castro
and F. S. Teixeira.
07/03/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Ana Moura Santos, Instituto Superior Técnico
An Operator Approach for an Oblique Derivative
Boundary-Transmission Problem
We consider a boundary-transmission problem for the Helmholtz
equation in a Bessel potential space setting. The boundary is a
strip of infinite extent and certain boundary conditions are
assumed on it in the form of oblique derivatives. The problem has
an interpretation within the context of diffraction theory and we
discuss the relevance of oblique derivatives boundary conditions.
Operator theoretical methods are used to deal with the problem and,
consequently, several convolution type operators are constructed
and "associated" to the problem. We also compare these results with
the previous construction of operators in the half-plane case. At
the end, the well-posedness of the problem is shown for orders of
the Bessel potential space near to that of the finite energy norm.
28/02/2003, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro
A Further Progress in the Theory of Lebesgue Spaces with
VariableExponent: Singular Integral Equations and Sobolev Theorem
forPotentials
The talk provides a discussion of recent results for the
generalized Lebesgue spaces with variable exponent \(p(x)\) (GLSVE)
including the criterion for the weighted singular operator (with a
power weight) to be bounded in such spaces. This result is applied
to "localize" the Gohberg-Krupnik criterion of Fredholmness of
singular integral operators in such spaces on Lyapunov curves. Some
abstract Banach space reformulation of the Gohberg-Krupnik scheme
of investigation of Fredholmness is given, from which the result
for GLSVE, in particular follows due to the boundedness criterion
for the weighted singular operator. Another new result for GLSVE
presented is the Sobolev theorem for potentials over the Euclidean
space, in which the "new word" is a possibility to consider the
variable exponent \(p(x)\) not necessarily constant at infinity.
However, the "payment" for this possibility is an additional power
weight fixed to infinity, which turns to be equal to \(1\) in the
traditional case \(p(x)=\operatorname{const}\).
21/02/2003, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Instituto Superior Técnico, U.T. Lisboa
Necessary Conditions for Fredholmness of Singular Integral Operators with PC Coefficients on Banach Function Spaces
We prove necessary conditions for Fredholmness of singular integraloperators with coefficients in the Banach algebra of piecewise continuous functions on weighted Banach function spaces. These conditions are formulated in terms of indices of a submultiplicative function associated with local properties of the space, of the curve, and of the weight. As an example, we consider the Musielak-Orlicz space \(L_{p(t)}\) (the Lebesgue space with variable exponent). In this example the above mentioned indices coincide with \(1/p(t)\) and \(p(t)/[p(t)-1]\) at each point (for nice curves and weight \(w=1\)). Our results give a natural generalization of the necessity part of the Gohberg-Krupnikcondition (for nice curves) as well as, the Boettcher - Yu. Karlovich condition (for general Carleson curves). So, we give a partial answer on the question raised by S. Samko on the roundtable on December 17, 2002.
17/12/2002, 15:15 — 16:15 — Sala P5, Pavilhão de Matemática
Ilya Spitkovsky, College of William and Mary, Williamsburg, VA, USA
Factorization of Almost Periodic Matrix Functions and its
Applications
We will give an overview of the current state of the spectral
theory of Toeplitz operators with matrix semi almost periodic
symbol. The role of the factorization problem for purely almost
periodic (in Bohr sense) matrix functions will be explained.
Existence of factorization will be discussed along with algorithms
of its actual construction. Applications include (but are not
limited to) convolution type equations on finite intervals.
17/12/2002, 14:00 — 15:00 — Sala P5, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, México
Pseudodifferential Operators and Their Applications
The talk is devoted to a symbol calculus for Banach algebras of
pseudodifferential operators (PDO's) with slowly oscillating data
and their applications to an interpolation theorem for singular
integral operators on weighted Lebesgue spaces and to the
calculation of local spectra for singular integral operators with
shifts and slowly oscillating data on Lebesgue spaces with
Muckenhoupt weights over Carleson curves. The study is based on the
Carleson-Hunt theorem on almost everywhere convergence and its
applications to the boundedness of PDO's, on the techniques of
oscillatory integrals, and on the theory of Mellin PDO's with
slowly oscillating symbols.
06/12/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Instituto Superior Técnico, U.T.L.
A Class of Singular Integral Operators with Flip and Unbounded
Coefficients on Rearrangement-Invariant Spaces
We prove Fredholm criteria for singular integral operators of the
form , where and are the Riesz projections,
is the flip operator, on a reflexive rearrangement-invariant
space with nontrivial Boyd indices over the unit circle. We assume
a priori that a is bounded, but may be unbounded. The function
belongs to a class of, in general, unbounded functions that
relates to the Douglas algebra . This result is new
even for Lebesgue spaces. It refines and generalizes some results
of Kravchenko, Lebre, Litvinchuk, and Teixeira published in the
Mathematische Nachrichten in 1995.
29/11/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro
Periodization of Two-Dimensional Fractional Riesz Operators
We consider the periodization of the Riesz fractional integrals
(Riesz potentials) of two variables and show that already in this
case we come across different effects, depending on whether we use
the repeated periodization, first in one variable, and afterwards
in another one, or the so called double periodization. We show that
the naturally introduced doubly-periodic Weyl-Riesz kernel of order
less than 2, in general coincides with the periodization of the
Riesz kernel, the repeated periodization being possible for all
orders , while the double one is applicable only for orders less
than 1. This is obtained as a realization of a certain general
scheme of periodization, both repeated and double versions. We
prove statements on coincidence of the corresponding periodic and
non-periodic convolutions and give an application to the case of
the Riesz kernel.
15/11/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro
Sonine Integral Equations of the First Kind
An integral equation of the first kind \[ K\phi(x):\equiv\int_{-\infty}^x k(x-t)\phi(t)\, dt = f(x)\] with a locally integrable kernel $k(x)\in L_1^{loc}(\mathbb{R}_+^1)$ is called Sonine equation if there exists another locally integrable kernel $\ell(x)$ such that \[\int_0^x k(x-t)\ell(t)\, dt \equiv 1\] (locally integrable divisors of the unit, with respect to the operation of convolution). In this case the formal solution of the equation is $\phi(x)=\frac{d}{dx}\int_0^x \ell(x-t)f(t)\, dt$. However, this inversion operator is formal: it does not work, for example, for solutions in the spaces $X=L_p(\mathbb{R}^1)$ and is not defined on the whole range $K(X)$.
We develop many properties of Sonine kernels which allow us — in a very general case — to construct the real inverse operator, within the framework of the spaces $X=L_p(\mathbb{R}^1)$, in Marchaud form:\[K^{-1}f(x)= \int_0^\infty \ell'(t)[f(x-t)-f(x)]\, dt\] with the interpretation of the convergence of this “hypersingular” integral in $L_p$-norm. The description of the range $K(X)$ is given; it requires already the language of Orlicz spaces even in the case when $X$ is the Lebesgue space $L_p(\mathbb{R}^1)$.
Some examples are considered.
08/11/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Cristina Câmara, Instituto Superior Técnico, U.T. Lisboa
Factorização de Funções Matriciais e Problemas de Riemann-Hilbert numa Superfície de Riemann
O estudo da factorização de Wiener-Hopf de funções matriciais $G$ de tipo $2\times 2$, em que os factores bem como os seus inversos pertencem a espaços apropriados de funções analíticas, pode ser reduzido à resolução de problemas de Riemann-Hilbert matriciais, relativos ao contorno $C$ onde a função $G$ está definida.
Nesta apresentação mostra-se a equivalência desses problemas matriciais, relativos a um contorno no plano, a um problema Riemann-Hilbert escalar relativo a um contorno numa superfície de Riemann e delineiam-se métodos para a sua resolução.
18/10/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Luís Filipe Pinheiro de Castro, Universidade de Aveiro
Existência, Unicidade e Diferenciabilidade de Soluções de EquaçõesIntegrais Não Lineares por via de Teoremas de Ponto Fixo
Consideram-se equações funcionais não lineares com representação integral que englobam, entre outras, as equações integrais de Urysohn, Hammerstein e pantográficas. É apresentado o método dos operadores de Picard para, através de teoremas de ponto fixo, se provar existência, unicidade, continuidade, bem como diferenciabilidade à Fréchet, para as soluções das equações em estudo.
11/10/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Instituto Superior Técnico, U.T.L.
Algebras of Singular Integral Operators on Rearrangement-InvariantSpaces and Nikolski Ideals
We construct a presymbol for the Banach algebra $\operatorname{alg}(A, S)$ generated by the Cauchy singular integral operator $S$ and the operators of multiplication by functions in a Banach subalgebra $A$ of essentially bounded functions. This presymbol mapping is a homomorphism of $\operatorname{alg}(A,S)$ onto $A+A$ whose kernel coincides with the commutator ideal of $\operatorname{alg}(A,S)$. In terms of the presymbol, necessary conditions for Fredholmness of an operator in $\operatorname{alg}(A,S)$ are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.