We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.

We study the singular integral operator

$${{T}_{\Omega ,\alpha }}f\left( x \right)\,=\,\text{p}\text{.v}\text{.}\,{{\int }_{{{R}^{n}}}}\,b\left( \left| y \right| \right)\Omega \left( {{y}'} \right){{\left| y \right|}^{-n-\alpha }}\,f\left( x\,-\,y \right)\,dy,$$

defined on all test functions $f$, where $b$ is a bounded function, $\alpha \ge 0,\,\Omega \left( {{y}'} \right)$ is an integrable function on the unit sphere ${{S}^{n-1}}$ satisfying certain cancellation conditions. We prove that, for $1\,<\,p\,<\infty$, ${{T}_{\Omega ,\alpha }}$ extends to a bounded operator from the Sobolev space $L_{\alpha }^{p}$ to the Lebesgue space ${{L}^{p}}$ with $\Omega$ being a distribution in the Hardy space ${{H}^{q}}\left( {{S}^{n-1}} \right)$ where $q=\frac{n-1}{n-1+\alpha }$. The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for ${{T}_{\Omega ,\alpha }}$ on the Hardy spaces, as well as the boundedness for the truncated maximal operator $T_{\Omega ,m}^{*}$.

Le problème de Jung-Nagata $\left( cf.\,\left[ \text{J} \right],\,\left[ \text{N} \right] \right)$ consiste à savoir s'il existe des automorphismes de $k\left[ x,\,y,\,z \right]$ qui ne sont pas modérés. Nous proposons une approche nouvelle de cette question, fondée sur l'utilisation de la théorie des automates et du polygone de Newton. Cette approche permet notamment de généraliser de façon significative les résultats de $\left[ \text{A} \right]$.

In this paper we consider linear systems of $\mathbb{P}{{}^{2}}$ with all but one of the base points of multiplicity 5. We give an explicit way to evaluate the dimensions of such systems.

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szegö polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szegö polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.

A Hopf surface is the quotient of the complex surface ${{\mathbb{C}}^{2}}\,\backslash \,\left\{ 0 \right\}$ by an infinite cyclic group of dilations of ${{\mathbb{C}}^{2}}$. In this paper, we study the moduli spaces ${{\mathcal{M}}^{n}}$ of stable $\text{SL}\left( 2,\,\mathbb{C} \right)$-bundles on a Hopf surface $\mathcal{H}$, from the point of view of symplectic geometry. An important point is that the surface $\mathcal{H}$ is an elliptic fibration, which implies that a vector bundle on $\mathcal{H}$ can be considered as a family of vector bundles over an elliptic curve. We define a map $G:\,{{\mathcal{M}}^{n}}\,\to \,{{\mathbb{P}}^{2n+1}}$ that associates to every bundle on $\mathcal{H}$ a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map $G$ is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on ${{\mathcal{M}}^{n}}$. We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

Given a $p$-dimensional oriented foliation of an $n$-dimensional compact manifold ${{M}^{n}}$ and a transversal invariant measure $\tau$, Sullivan has defined an element of ${{H}_{p}}\left( {{M}^{n}},\,R \right)$. This generalized the notion of a $\mu$-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure $\mu$. In this one-dimensional case there was a natural 1–1 correspondence between transversal invariant measures $\tau$ and invariant measures $\mu$ when one had a smooth flow without stationary points.

For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

We describe the irrational pencils on surfaces of general type with ${{p}_{g}}=q=2$.