A note on multilinear fractional integrals

Shi and Tao[6] studied the boundedness of multilinear fractional integrals introduced by Kenig and Stein[3] on product of weighted L ^p -spaces, and got some results. We give some remarks with respect to their results and correct some mistakes. We also consider another multilinear fractional integral introduced by Grafakos[2].     

Values of Gaussian hypergeometric series

Let be prime and let be the finite field with elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions

where and respectively are the quadratic and trivial characters of For all but finitely many rational numbers there exist two elliptic curves and for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes for which is zero; however if or , then the set of such primes has density zero. In contrast, if or , then there are only finitely many primes for which Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating over every

     

一类Marcinkiewicz积分算子的双权有界性

本文研究了Marcinkiewicz积分μΩ的有界性的问题,借助Ap权的性质,得到了相应与Littlewood-Paley g-函数的Marcinkiewica积分算子μΩ的(Lp(v),Lp(u))有界性,并且证明了相应于Littlewood-Paley gλ*-函数和Lusin面积积分的Marcinkiewicz积分μ*Ω,λ,μΩ,S的(Lp(v),Lp(u))有界性.     

WEIGHTED L^p BOUNDEDNESS FOR MULTILINEAR FRACTIONAL INTEGRAL ON PRODUCT SPACES

For 0 〈 α 〈 mn and nonnegative integers n ≥ 2, m≥ 1, the multilinear fractional integral is defined bywhere →y= （y1, Y2,…, ym） and 7 denotes the m-tuple （f1, f2,…, fm）. In this note, the one- weighted and two-weighted boundedness on Lp （JRn） space for multilinear fractional integral operator I（am） and the fractional multi-sublinear maximal operator Mα（m） are established re- spectively. The authors also obtain two-weighted weak type estimate for the operator Mα（m）.     

The simulation of nanoscale sputter depth profiles using molecular dynamics

The feasibility of using molecular dynamics (MD) for simulation of a nanoscale sputter depth profile experiment is examined for the idealised case of depth profiles of individual atomic layers in a Cu(1 0 0) target. Issues relating to the extraction of depth profile information from MD simulations are discussed in detail. The simulations examine the sputter erosion of static and azimuthally-rotated Cu(1 0 0) targets produced by 3 keV Ar projectiles incident at 25° from the surface. The simulated projectile fluence extends to 5 × 10¹⁵ cm⁻², and the mean value of the sputter depth, z, amounts to 8 Cu(1 0 0) monolayers (ML) or 15 Å. The simulations directly supply progressive layer erosion profiles (curves that depict the extent of sputter erosion of each atomic layer vs. total sputter depth). A fitting method is then used to extract smooth depth profiles for each atomic layer from these predicted erosion profiles. The depth profile characteristics (height, width, shift) for the first 10 layers of the target show a pronounced dependence on layer depth.     

First-principles calculations of the electronic and optical properties of In6S7 compound

The electronic structure and the optical properties of In₆S₇ crystal are calculated by the first-principles full-potential linearized augmented plane wave method (FP-LAPW) using density functional theory (DFT) in its generalized gradient approximation (GGA). The calculated band structure shows that the In₆S₇ is a semiconductor with a direct band gap in good agreement with experimental studies. Furthermore, the dielectric tensor and the optical properties, such as absorption coefficient, refractive index, extinction coefficient, energy-loss spectrum and reflectivity, are derived and analyzed in the study.     

Density functional approach to study 5f electron behavior and electric field gradient in NpRh3

We have presented results on the electronic structure and the electric field gradient (EFG) of NpRh₃ within a framework of density functional theory within the local density approximation (LDA) and generalized gradient approximation (GGA) with and without spin-orbit coupling (SOC). The electronic density of states (DOS) shows that Np-f to Rh-d hybridization leads to a narrow band 5f-electron. Using the GGA and spinpolarized calculations, the calculated EFG shows that the dominant contribution to EFG comes from electrons with strong p-character.     

Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincaré-type inequalities for Hörmander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inqualities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on ?. Some very usefull properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38].The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (?,δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.     

On the initiation of chemical reactions by electronic excitations in molecular solids

In this article we briefly review the results of theoretical simulations for the initiation of chemistry processes in high-explosive crystals from a solid-state-physics viewpoint. We analyze the possibility of initiation of chemical reactions from excited electronic states. In other words, we look for conditions that facilitate electronic excitations in the crystal. Specifically, we describe modifications to the electronic structure of RDX (cyclotrimethylene trinitramine) induced by lattice defects and by a shock wave traversing the solid. Our approach is based on ab initio Hartree–Fock band-structure calculations with electronic correlation corrections. An excitonic mechanism and a hole model, suggested earlier, are discussed in connection with the most recent experimental and theoretical advances in ultrafast optical techniques. We also consider here possible new avenues in the development of detonation theory. Received: 3 December 2001 / Accepted: 9 July 2002 / Published online: 25 October 2002 RID="*" ID="*"Corresponding author. Fax: +1-301/744-4451, E-mail: KuklaMM@ih.navy.mil     

Local properties of the electronic structure of cubic SrTiO3, BaTiO3 and PbTiO3 crystals, analysed using Wannier-type atomic functions

The electronic structure of the cubic perovskites SrTiO₃, BaTiO₃ and PbTiO₃ is calculated by Hartree-Fock and density functional theory methods. Wannier-type atomic orbitals (WTAOs) are obtained from symmetrized combinations of Bloch states of some occupied and vacant bands by a variational method. Population analysis, based on the WTAOs, attributes a mixed ionic-covalent type of bonding with partly covalent Ti-O bonds to the perovskites under study. The atomic charges thus calculated are then compared to those obtained by the traditional Mulliken population analysis.