Functional Analysis and Applications Seminar 

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 $L_q$-spaces we will use tools from harmonic analysis and a special maximal operator reflecting paths of fluid particles past or around the obstacle.

We introduce and study the concept of the Banach-Saks index. It is motivated by the Banach-Saks and $p$-Banach-Saks property. The Banach-Saks index is related with main geometrical characteristics of Banach functional spaces ( $p$-convexity and $q$-concavity, Rademacher type, Boyd indices). We study operators related with the Banach-Saks index.

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.

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.

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 $H^{\infty }$);
• Weak generators in quotient algebras of $H^{\infty }$;
• Operator corona problem and approximation property.

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.

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.

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.

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).

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.