Compactness of Riesz transform commutator associated with Bessel operators

Let λ > 0 and

$${\Delta _\lambda }: = - \frac{{{d^2}}}{{d{x^2}}} - \frac{{2\lambda }}{x}\frac{d}{{dx}}$$

be the Bessel operator on R₊:= (0,∞). We first introduce and obtain an equivalent characterization of CMO(R₊, x^2λdx). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function b ∈ BMO(R⁺, x^2λdx) is in CMO(R⁺, x^2λdx) if and only if the Riesz transform commutator xxxx is compact on L^p(R⁺, x^2λdx) for all p ∈ (1,∞).

     

New extragradient-like algorithms for strongly pseudomonotone variational inequalities

The paper considers two extragradient-like algorithms for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The projection method is used to design the algorithms which can be computed more easily than the regularized method. The construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the modulus of strong pseudomonotonicity and the Lipschitz constant of the cost operator. Instead of that, the algorithms use variable stepsize sequences which are diminishing and non-summable. The numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with those of several previously known algorithms.     

Two-Weighted Inequalities for Hausdorff Operators in Herz-Type Hardy Spaces

Mathematical Notes - In this paper, we prove the boundedness of matrix Hausdorff operators and rough Hausdorff operators in the two weighted Herz-type Hardy spaces associated with both power...     

Weighted Norm Inequalities for Commutators of BMO Functions and Singular Integral Operators with Non-Smooth Kernels

The aim of this paper is to establish a sufficient condition for certain weighted norm inequalities for singular integral operators with non-smooth kernels and for the commutators of these singular integrals with BMO functions. Our condition is applicable to various singular integral operators, such as the second derivatives of Green operators associated with Dirichlet and Neumann problems on convex domains, the spectral multipliers of non-negative self-adjoint operators with Gaussian upper bounds, and the Riesz transforms associated with magnetic Schrödinger operators.     

Validity of the Lattice-Parameter Vegard-Rule in Cd1-xZnxTe Solid Solutions

Cd_1-xZn_xTc crystals of different composition (0 ⩽ x ⩽ 1) were grown by the vertical Bridgman method and by synthesis in Te excess. After careful surface preparation of crystal slices, their Zn:Cd ratios were ascertained by wavelength-dispersive X-ray spectroscopy (WDXS), and the lattice parameters of the same slice regions were measured by X-ray diffractometry. The Vegard rule concerning the linear lattice-parameter dependence on composition is – in contrast with earlier literature data – exactly valid within the limits of error (Δx ⩽ ±0.01 and Δa/a ⩽ 3.7. 10⁻⁴) and follows the equation a(x) = (0.64822 - 0.03792x) nm. After annealing some slices at different temperatures and controlled partial pressure conditions in ordcr to find possible phase separations or cation ordering effects, neither the lattice parameters were changed nor additonal interferences were found. A published occurrence of a rhomhohedral phase could not be confirmed by means of powder diffraction analysis. The results are discussed in relation to own electron diffraction investigations and to EXAFS literature data.     

Liouville-type theorems for a nonlinear fractional Choquard equation

In this paper, we are concerned with the fractional Choquard equation on the whole space ${\mathbb{R}}^N$ $${(-\mathrm{\Delta })}^{s}u=\left(\frac{1}{{|x|}^{N-2s}}\ast u^p\right)u^{p-1}$$ with , $N>2s$ and $p\in \mathbb{R}$. We first prove that the equation does not possess any positive solution for $p\le 1$. When $p>1$, we establish a Liouville type theorem saying that if then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.