The finite Hilbert transform and weighted Sobolev spaces

The boundedness of the finite Hilbert transform operator on certain weighted L_p spaces is well known. We extend this result to give the boundedness of that operator on certain weighted Sobolev spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hodge–Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media

We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains Ω??^N filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rank q belonging to the weighted L²‐space L_s^{2, q}(Ω), s∈?, into irrotational and solenoidal q‐forms. These decompositions are essential tools, for example, in electro‐magnetic theory for exterior domains. To the best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results to the classical framework of vector analysis N=3 and q=1, 2. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonstationary Stokes system in weighted Sobolev spaces

We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??³ with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L_{2, ?µ}(Ω^T), initial velocity v₀∈H(Ω), µ∈?₊\? there exist velocity v∈H(Ω^T) and the pressure p, ?p∈L_{2, ?µ}(Ω^T) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:

• 1. the slip boundary condition and
• 2. the Helmholtz–Weyl decomposition.

Copyright © 2010 John Wiley & Sons, Ltd.     

Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness

This paper is a continuation of work of the author and joint work with Winfried Sickel. Here we shall investigate the asymptotic behaviour of Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness into Lebesgue spaces.     

Atomic decomposition for preduals of Sobolev multiplier spaces

We characterize the preduals of Sobolev multiplier spaces by means of atomic decomposition. It will be shown that the vanishing integral condition plays a main role in the atomic decomposition. Certain subspaces of the preduals which consist of functions with vanishing integrals are also studied.     

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

The IVP for the Benjamin-Ono equation in weighted Sobolev spaces

We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces , s∈R, s?1 and s?r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp.     

A duality principle in weighted Sobolev spaces on the real line

An embedding inequality of Sobolev type is characterized in the paper with help of a duality principle and boundedness criteria for the Hardy–Steklov integral operator in weighted Lebesgue spaces.     

Non-homogeneous Dirichlet boundary value problems in weighted Sobolev spaces

The paper presents existence and multiplicity results for non-linear boundary value problems on possibly non-smooth and unbounded domains under possibly non-homogeneous Dirichlet boundary conditions. We develop here an appropriate functional setting based on weighted Sobolev spaces. Our results are obtained by using global minimization and a minimax approach using a non-smooth critical point theory.     

Jumping nonlinearities and weighted Sobolev spaces

Working in a weighted Sobolev space, a new result involving jumping nonlinearities for a semilinear elliptic boundary value problem in a bounded domain in R^N is established. The nonlinear part of the equation is assumed to grow at most linearly and to be at resonance with the first eigenvalue of the linear part on the right. On the left, the nonlinearity crosses over (or jumps over) several higher eigenvalues. Existence is obtained through the use of infinite-dimensional critical point theory in the context of weighted Sobolev spaces and appears to be new even for the standard Dirichlet problem for the Laplacian.     

Nonstationary Stokes system in anisotropic Sobolev spaces

The nonstationary Stokes system with slip boundary conditions is considered in a bounded domain . We prove the existence and uniqueness of solutions to the problem in anisotropic Sobolev spaces . Thanks to the slip boundary conditions, the Stokes problem is transformed to the Poisson and the heat equation. In this way, difficult calculations that must be performed in considerations of boundary value problems for the Stokes system are avoided. This approach does not work for the Dirichlet and the Neumann boundary conditions. Because solvability of the Poisson and the heat equation is carried out by the regularizer technique, we have that σ > 3,α > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

Anisotropic equations in weighted Sobolev spaces of higher order

We consider the strongly nonlinear boundary value problem,

On the periodic KdV equation in weighted Sobolev spaces

We prove well-posedness results for the initial value problem of the periodic KdV equation as well as Kam type results in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential decay of Fourier coefficients, and ‘almost well posed’ in spaces with exponential decay of Fourier coefficients.     

Continuity and differentiability for weighted Sobolev spaces

Our aim in this paper is to discuss continuity and differentiability of functions in weighted Sobolev spaces in the limiting case of Sobolev's imbedding theorem.

     

Nonstationary Stokes system in Besov spaces

We examine the solvability in Besov spaces of an initial–boundary value problem for the nonstationary Stokes system with the slip boundary conditions. We prove the existence and uniqueness of solutions to the problem in a bounded domain . The existence is shown by localizing the system to interior and boundary subdomains of Ω. The localized Stokes system is transformed by the Helmholtz–Weyl decomposition to the heat and the Poisson equations, which are solved in the Besov spaces. Next, by the properties of the partition of unity and a perturbation argument, the existence is proved in domain Ω. Copyright © 2013 John Wiley & Sons, Ltd.     

Sharp conditions for the compactness of the Sobolev embedding on Musielak–Orlicz spaces

We prove the compactness of the Sobolev embedding for Musielak–Orlicz spaces by way of simple conditions on the Matuszewska index of the underlying space. We provide counterexamples to show the sharpness of our conditions.     

A simple characterization of weighted Sobolev spaces with bounded multiplication operator

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotone weights) such that the multiplication operator is bounded: it is bounded if and only if the support of μ₀ is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials.     

Poincaré's inequality and Sobolev spaces with monomial weights

In this paper, we use a weighted version of Poincaré's inequality to study density and extension properties of weighted Sobolev spaces over some open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$. Additionally, we study the specific case of monomial weights $w(x_1,\text{…},x_N)={\prod }_{i=1}^N{\left|x_i\right|}^{a_i},a_i\ge 0$, showing the validity of a weighted Poincaré inequality together with some embedding properties of the associated weighed Sobolev spaces.     

Elliptic equations with nonzero boundary conditions in weighted Sobolev spaces

We present weighted Sobolev spaces along with a trace theorem and an interpolation theorem for the spaces. Then we solve nonzero boundary value problems for elliptic equations in .