Complex cauchy problem in the class of analytic functions with integral metric

We study the complex Cauchy problem for a system of linear differential equations in the class of analytic functions with integral metric. In the case of a Hardy-Lebesgue type weighted L _p -space, we obtain necessary and sufficient conditions for the local solvability of the problem.     

On the bounded approximation property in Banach spaces

We prove that the kernel of a quotient operator from an L ₁-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case ? ₁-and Figiel, Johnson and Pe?czyński-case X* separable. Given a Banach space X, we show that if the kernel of a quotient map from some L ₁-space onto X has the BAP, then every kernel of every quotient map from any L ₁-space onto X has the BAP. The dual result for L _∞-spaces also holds: if for some L _∞-space E some quotient E/X has the BAP, then for every L _∞-space E every quotient E/X has the BAP.     

A new Lindelöf topological group

We show that the subsemigroup of the product of ω₁-many circles generated by the L-space constructed by J. Moore is again an L-space. This leads to a new example of a Lindelöf topological group. The question whether all finite powers of this group are Lindelöf remains open.     

On Weighted Estimates of High-Order Riesz-Bessel Transformations Generated by the Generalized Shift Operator

In this paper, we establish sufficient conditions on weights which ensure that high-order Riesz-Bessel transformations generated by the generalized shift operator act boundedly from one weighted Lp-space into another.     

Fixräume regulärer operatoren und ein Korovkin-Satz

S. J. Bernau has introduced the notion of an exchange subspace of an L_p-space and has shown that the range of a contractive linear projection on an L_p-space (1 ? p < ∞, p ≠ 2) is an exchange subspace. In the present paper we define this notion for real Banach lattices with order continuous norm and prove among other things that fixed spaces of special regular operators on these spaces are exchange subspaces. As application we give a Korovkin theorem for sequences of contractions on real Banach lattices with an uniformly monotone norm.     

A generalised Christoffel-Darboux formula and reproducing kernels in spaces of polymonogenic and polyharmonic functions

Let ?_x be the Dirac operator in R^m+1, let t ∈ N and let K_t(x,y) be the Bergman kernel for the space L ₂M_t of square integrable polymonogenic functions of order t in the unit ball B(1) of R^m+1. Then expressions for K_t(x,y) are derived using a generalised Christoffel-Darboux formula related to the orthogonal basis of generalised Gegenbauer polynomials in the weighted L₂-space L₂(B(1);(1 + x²⁾α), α > ?1.     

Geometric description of L 1-Spaces

We describe strongly facially symmetric spaces which are isometrically isomorphic to L ₁-space.     

Generalized boundary value problems for semilinear elliptic equations in weighted function spaces

In a weighted L ₁-space, we prove the solvability of a boundary value problem for a semilinear elliptic equation of order 2m in a bounded domain for the case in which generalized functions with strong power-law singularities at isolated points and with finite-order singularities on the entire boundary are given on the boundary.     

On the periodic Wigner-Poisson-Fokker-Planck system

This paper is concerned with the existence and uniqueness analysis of global classical solutions of a diffusive quantum evolution equation with nonlinear coupling to the Poisson equation. The main technical difficulty in the existence proof is to show that the quantum Fokker-Planck term is a semigroup-generator in a weighted L²-space. The potential term is then a Lipschitz perturbation of it.     

On the Transverse Vibrations of a String Segment Between Two Supports Moving Axially Along a String of Infinite Length

This paper deals with the equation of motion of a string with time-varying boundary conditions. This equation can be transformed to a partial differential equation (PDE) with non-varying boundary conditions. During the solution process, the inverse of the time derivative operator is needed. To ensure the existence of a solution, the time derivative is generalized to a weighted time derivative on a weighted L₂-space. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A weak invertibility criterion in the weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces of holomorphic functions in the ball

We obtain a necessary and sufficient condition on a weight function for every nowhere vanishing holomorphic function in the unit ball in the weighted L ^p -space to be weakly invertible in the corresponding L ^q -space for all q < p.     

Spectrum topology of a residuated lattice

In this paper, given a non-commutative residuated lattice L, a topological space is constructed using certain fuzzy subsets of L. Indeed, we show that the set of all prime fuzzy filters of a non-commutative residuated lattice L forms a topological space. Particularly, we show that this space is compact and a T ₀-space and its certain subspaces are Hausdorff spaces. Finally, we show that the set of all prime filters of L is also a Hausdorff space.     

On a classifying property of regular representations

We show that, for each connected compact Lie group G, the Hilbert G-space L ₂(G) and the Banach G-space C(G;?) classify the G-spaces.     

Is the linearized Boltzmann-Enskog operator dissipative?

In this paper, it is shown that the linearized Boltzmann-Enskog collision operator cannot be dissipative in the L₂-space setting contrarily to the linearized Boltzmann operator. Some estimates useful for the spectral theory are given.     

Generalized boundary values of solutions of quasilinear elliptic equations with linear principal part

We establish conditions for the nonlinear part of a quasilinear elliptic equation of order 2m with linear principal part under which a solution regular inside a domain and belonging to a certain weighted L ₁-space takes boundary values in the space of generalized functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1674–1688, December, 2007.     

A ∞ weight estimates for vector-valued commutators of multilinear fractional integral

In this paper, the authors get the Coifman type weighted estimates and weak weighted LlogL estimates for vector-valued generalized commutators of multilinear fractional integral with w ∈ A∞. Furthermore, both the boundedness of vector-valued multilinear frac- tional integral and the weak weighted LlogL estimates for vector-valued multilinear fractional integral are also obtained.     

Spectra of some block operator matrices and application to transport operators

In this paper a new concept for a 3×3 block operator matrix is studied on a Banach space. It is shown that, under certain conditions, it defines a closable operator and its essential spectra are determined. Application to transport operators in L₁-space is given.     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

The Bishop-Phelps-Bollobás theorem for operators

We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from ?₁ into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L₁(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.     

Representation of linear functionals on a class of normed Köthe spaces

Under the condition that $\text{L}{}_{\mathrm{\theta ,s}}{}^1$, (the set of singular functionals on a normed Köthe space L_θ) is an abstract L-space, it is proved in this paper that there exists a set of purely finitely additive measures $\text{M}$_θ such that $\text{L}{}_{\mathrm{\theta ,s}}{}^1$? holds. It follows that $\text{L}{}_{\mathrm{\theta ,s}}{}^1$ is an abstract L-space if and only if $\text{L}{}_{\mathrm{\theta ,s}}{}^1$ is Riesz isomorphic and isometric with a band in $\text{L}{}_{\mathrm{\infty ,s}}{}^1$.