Invariant integral and statistical equations of paraxial light beam transmission in free space

The statistical behavior of arbitrary paraxial light beams propagating in free space is investigated by using the Hermite-Gaussian expansion method and Fock’s representation. A series of equivalent Gaussian parameters for paraxial beam and the statistical equations for these parameters are presented. The optical transmission problem in quasi-far field region is studied. The so-called general Hermite-Gaussian beam is defined. Project partly supported by the National Hi-tech Inertial Confinement Fusion Committee, the Natural Science Foundation of Guangdong Province, the Postdoctoral Foundation of China and Guangdong Province.     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

Atomic decompositions of Banach-spacevalued martingales

Several theorems for atomic decompositions of Banach-space-valued martingales are proved. As their applications, the relationship among some martingale spaces such asH _α(X) and_ρ H _α in the case 0< α⩽ are studied. It is shown that there is a close connection between the results and the smoothness and convexity of the value spaces. Project supported by the National Natural Science Foundation of China (Grant No. 19771063).     

Composition Operators on the Bloch Space of Several Complex Variables

Abstract In this paper, we study the boundedness and compactness of composition operator C _φ on the Bloch space β(Ω), Ω being a bounded homogeneous domain. For Ω = B _n, we give the necessary and sufficient conditions for a composition operator C _φ to be compact on β(B _n) or β ₀(B _n). Supported by the National Natural Science Foundation and the National Education Committee Doctoral Foundation     

Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ _Ω ^ρ , μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ on BMO(ℝ ⁿ ) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog⁺ L(S ⁿ⁻¹). Here μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g _λ *-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ ⁿ ) or to a certain Campanato space, then [μ _Ω,λ ^*,ρ (f)]², [μ _Ω,S ^ρ (f)]² and [μ _Ω ^ρ (f)]² are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.     

General A-P iterative algorithm in shift-invariant spaces

A general A-P iterative algorithm in a shift-invariant space is presented. We use the algorithm to show reconstruction of signals from weighted samples and also show that the general improved algorithm has better convergence rate than the existing one. An explicit estimate for a guaranteed rate of convergence is given.     

The Dissipative Quasi-geostrophic Equation in Weak Morrey Spaces

The author studies the Cauchy problem of the dissipative quasi-geostrophic equation in weak Morrey spaces. The global well-posedness is established for any small initial data in the weak space Mp^＊,γ（R^n）, with 1〈p〈∞and A = n-（2α-1）p, and for a small external force in a time-weighted weak Morrey space.     

Disk-like Tiles Derived from Complex Bases

For each positive integer k, the radix representation of the complex numbers in the base –k+i gives rise to a lattice self-affine tile T _k in the plane, which consists of all the complex numbers that can be expressed in the form ∑ _j≥1 d _j (–k+i)^–j , where d _j ∈{0, 1, 2, ...,k ²}. We prove that T _k is homeomorphic to the closed unit disk {z∈C:∣z∣ ≤ 1} if and only if k ≠ 2. The first author is supported by Youth Project of Tianyuan Foundation (10226031) and Zhongshan University Promotion Foundation for Young Teachers (34100-1131206); the second author is supported by National Science Foundation (10041005) and Guangdong Province Science Foundation (011221)     

Compactness of Composition Operators from the Bloch Space B to QK Spaces

Suppose that φ is an analytic self-map of the unit disk Δ. We consider compactness of the composition operator Cφ from the Bloch space B into the spaces QK defined by a nonnegative, nondecreasing function K（r） for 0 ≤ r 〈 Cφ. Our compactness condition depends only on Φ which can be considered as a slight improvement of the known results. The compactness of Cφ from the Dirichlet space D into the spaces QK is also investigated,     

Multipliers on Dirichlet Type Spaces

In this paper, we characterize the pointwise multiplier space M (D _τ, D _μ) of Dirichlet type spaces in the unit ball of C ⁿ for the values of τ, μ in three cases: (i) τ < 0,μ < 0, (ii) τ < μ, (iii) τ≥μ, τ > n, and construct two functions to show that M(D _τ) ⊂D _τ properly if τ≤n and M(D _τ) ⊂M(D _μ) properly if τ > μ and τ > n− 1. Supported by the National Natural Science Foundation of China and the National Education Committee Doctoral Foundation of China     

On the equivalence of extremal Teichmüller mapping

Let T(Δ) and B(Δ) be the Teichmüller space and the infinitesimal Teichmüller space of the unit disk Δ respectively. In this paper, we show that [ν] _B(Δ) being an infinitesimal Strebel point does not imply that [ν] _T(Δ) is a Strebel point, vice versa. As an application of our results, problems proposed by Yao are solved. This work was supported by the National Natural Science Foundation of China (Grant No. 10571028)     

Hypersingular parameterized Marcinkiewicz integrals with variable kernels on Sobolev and Hardy-Sobolev spaces

Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.     

On structure of small contractions of odd dimensional projective varieties

Let X be a smooth projective variety of dimension 2k-1 (k≥3) over the complex number field. Assume that fR: X→Y is a small contraction such that every irreducible component Ei of the exceptional locus of fR is a smooth subvariety of dimension k. It is shown that each Ei is isomorphic to the k-dimensional projective space Pk, the k-dimensional hyperquadric surface Qk in Pk 1, or a linear Pk-1-bundle over a smooth curve.     

The regularity of the surface of the Gauss curvature 0≤K ∈ C 0 ∞

Under the mild conditions, it is proved that the convex surface is global C^1.1, with the given Gaussian curvature 0≤K ∈ C ₀ ^∞ and the given boundary curve. Examples are given to show that the regularity is optimal. Project supported by the Doctoral Funds of China and the National Natural Science Foundation of China (Grant No. 19771009).     

Basic Results of Gabor Frame on Local Fields

Basic facts for Gabor frame {Eu(m)bTu(n)ag}m,n∈p on local field are investigated. Accurately, that the canonical dual of frame {Eu(m)bTu(n)ag}m,n∈p also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu(m)bTu(n)ag}m,n∈p to be a frame for L2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L2(K) are established. An example is finally given.     

Further generalization of Fermat’s principle

From Maxwell’ s equations for electromagnetic fields, time-averaged energy flow density vector of stable monochromatic linearly polarized light in an isotropic insulative nonmagnetic medium is deduced. By the introduction of time-averaged energy flow density rays and the definition of new generalized refractive indexn _G1, Fermat’s principle of geometric optics is further generalized and its application conditions are discussed. The generalized Fermat' s principle can be used to describe stable transmission of light in a medium with variable refractive index. The necessary and sufficient conditions of a nondivergent and nonfocusing light beam are derived from this Fermat’s principle. Project supported by the National Natural Science Foundation of China (Grant No. 69789801) and Guangdong Provincial Natural Science Foundation of China.     

Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.     

The Willmore functional and the containment problem in R4

Let ∑ be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional f∑ H2dσ. This bound is an invariant involving the area of ∑, the volume and Minkowski quermassintegrals of the convex body that ∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

The Hausdorff Centred measure of the symmetry Cantor sets

Let 0＜λ≤1/3,K (λ) be the attractor of an iterated function system { ψ1,ψ2 } on the line, where ψ1 (x ) =λx, ψ2(x)=1-λ+λx, x∈ [0,1]. We call K (λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K (λ).     

A rough hypersingular integral operator with an oscillating factor on function space

The singular integral operator Tα,βf(x) = p.v.∫Rnei|y|-βΩ(y') /|y|n αf(x-y)dy,defined for all test functions f is studied, where Ω(y') is a distribution on the unit sphere Sn-1 satisfying certain cancellation condition. It is proved that Tα,β is a bounded operator from the Triebel-Lizorkin space Fs,qp to the Triebel-Lizorkin space Fs γ,qp, provided that Ω(y') is a distribution in the Hardy space Hr(Sn-1) with r = (n - 1)/(n - 1 γ).