Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.     

Hitting Half-spaces by Bessel-Brownian Diffusions

The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in ℝ ^n + 1, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ) ^α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process.     

Hamel‐isomorphic images of the unit ball

In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ?² and ?. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non‐measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non‐measurable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Many-body hydrodynamic interactions in charge-stabilized suspensions

In this joint experimental-theoretical work we study hydrodynamic interaction effects in dense suspensions of charged colloidal spheres. Using x-ray photon correlation spectroscopy we have determined the hydrodynamic function H(q), for a varying range of electrosteric repulsion. We show that H(q) can be quantitatively described by means of a novel Stokesian dynamics simulation method for charged Brownian spheres, and by a modification of a many-body theory developed originally by Beenakker and Mazur. Very importantly, we can explain the behavior of H(q) for strongly correlated particles without resorting to the controversial concept of hydrodynamic screening, as was attempted in earlier work by Riese [Phys. Rev. Lett. 85, 5460 (2000)].     

Isosceles-orthogonality preserving property and its stability

For real normed spaces, we consider the class of linear operators, preserving approximately the relation of isosceles-orthogonality. We show some general properties of such mappings. Next, we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one. In regard to this, we generalize some results obtained earlier for inner product spaces with standard orthogonality relation.     

Near Infrared Studies of Polymer Blends

Abstract

The dependence of absorption in the near-infrared region of cellulose acetate-polyaniline/camphorsulfonic acid blends on their conductivity and homogeneity has been studied. Thin films (10 to 100 μm) of the blends containing various amounts of protonated polyaniline show very high absorption in the NIR region. The absorption increases with increasing of conductivity and homogeneity of the sample.     

Towards uniqueness of degenerate axially symmetric Killing horizon

For near horizon geometry we examine the linearized equations around extremal Kerr horizon (which is a unique axially symmetric near horizon geometry) and give some arguments towards stability of this horizon with respect to generic (non-symmetric) linear perturbation of near horizon geometry. The result is also applicable for other situations like Kundt’s class spacetimes or isolated horizons.     

Laser Raman Investigation of Solid State Reactions

Abstract

Traditionally, molecular ions have been studied by mass spectrometers or ion-counting techniques. The great sensitivity and versatility of these techniques are clearly attested to by the vast amount of mass spectrometric literature. However, it is equally true that mass spectroscopy has its limitations; basically, it provides one only with a charge-to-mass ratio for an ion. Obviously, all quantum state information is lost, particularly the vibrational and rotational, and usually even the electronic distributions of the ions. Similarly, no structural information such as bond lengths and angles is obtained. Indeed, in some cases the information obtained is so slight that one cannot even write a structural formula for the ion or distinguish between different chemical isomers.     

Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

Let $\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )$ , be the multivariate Hankel transform, where $E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})$ , with dν(λ)=λ ^2α dλ, α=(α ₁,…,α _d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator $\mathcal{H}(m\mathcal{H} f)$ is bounded on L ^p (dν) and of weak-type (1,1), or bounded on the Hardy space H ¹((0,∞) ^d ,dν) in the sense of Coifman-Weiss.