Functional Analysis and Applications Seminar 

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.

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.

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.

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.

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.

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.

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.

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

1. operator relations,
2. a new factorization approach for convolution type operators with symmetry, and
3. 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.

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.

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 \(p(x)=\operatorname{const}\).

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.

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.

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.

We prove Fredholm criteria for singular integral operators of the form $P+aQ+bUQ$, where $P$ and $Q$ are the Riesz projections, $U$ 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 $b$ may be unbounded. The function $b$ belongs to a class of, in general, unbounded functions that relates to the Douglas algebra $H^{\mathrm{\infty }}+C$. 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.

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.

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.

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.

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.

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.