15 April 2013, Volume 29 Issue 4

    

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  Three-space Problems for the Bounded Compact Approximation Property

  Dong Yang CHEN, Ben Tuo ZHENG

  Acta Mathematica Sinica. 2013, 29(4): 625-632. https://doi.org/10.1007/s10114-012-0661-7

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  In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L_∞ with the BCAP, then L_∞/X has the BCAP. We also show that X^* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y⊆X. Let M be a closed subspace of X such that M^⊥ is complemented in X^*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.
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  Essential Norms of Composition Operators between Weighted Bergman Spaces of the Unit Disc

  Luo LUO, Jing CHEN

  Acta Mathematica Sinica. 2013, 29(4): 633-638. https://doi.org/10.1007/s10114-012-0070-y

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  In this paper, we express the essential norms of composition operators between weighted Bergman spaces of the unit disc in terms of the generalized Nevanlinna counting function.
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  Reflected Backward Doubly Stochastic Differential Equations with Discontinuous Coefficients

  Zhi LI, Jiao Wan LUO

  Acta Mathematica Sinica. 2013, 29(4): 639-650. https://doi.org/10.1007/s10114-013-2074-7

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  In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) generator. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.
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  A Microscopic Convexity Principle for Spacetime Convex Solutions of Fully Nonlinear Parabolic Equations

  Chuan Qiang CHEN, Bo Wen HU

  Acta Mathematica Sinica. 2013, 29(4): 651-674. https://doi.org/10.1007/s10114-012-1495-z

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  We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.
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  On Wiener-Poisson Type Multivalued Stochastic Differential Equations with Non-Lipschitz Coefficients

  Jing WU

  Acta Mathematica Sinica. 2013, 29(4): 675-690. https://doi.org/10.1007/s10114-012-1164-2

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  In this paper, we prove local uniqueness for multivalued stochastic differential equations with Poisson jumps. Then existence and uniqueness of global solutions is obtained under the conditions that the coefficients satisfy locally Lipschitz continuity and one-sided linear growth of b. Moreover, we also prove the Markov property of the solution and the existence of invariant measures for the corresponding transition semigroup.
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  Vector Cascade Algorithms with Infinitely Supported Masks in Weighted L₂-Spaces

  Jian Bin YANG

  Acta Mathematica Sinica. 2013, 29(4): 691-702. https://doi.org/10.1007/s10114-012-1295-5

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  In this paper, we shall study the solutions of functional equations of the form
    ,
  where Φ=(φ₁,...,φ_r)^T is an r×1 column vector of functions on the s-dimensional Euclidean space, a:= (a(α))α∈Z^s is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s×s integer matrix such that lim_n→∞M^-n=0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function φ_j, j=1, ..., r, belonging to L₂(R^s) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L₂ spaces. The vector cascade operator Q_a,M associated with mask a and matrix M is defined by
  
  The iterative scheme (Q_a,Mⁿ f)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L_2,μ(R^s))^r, the weighted L₂ space. Inspired by some ideas in [Jia, R. Q., Li, S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008-1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L₂(R^s))^r, then its limit function belongs to (L_2,μ(R^s))^r for some μ>0.
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  Non-recurrence of exp(z)/z

  Guo Ping ZHAN

  Acta Mathematica Sinica. 2013, 29(4): 703-716. https://doi.org/10.1007/s10114-012-1451-y

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  In this paper, we consider the dynamics of the map z→exp(z)/z on the punctured plane C^*=C\{0}. We show that for almost every point z∈C^*, the ω-limit set of z is equal to {0,∞}. In particular, the map is not recurrent.
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  On Inclusion of Permutation Modules

  Li Zhong WANG

  Acta Mathematica Sinica. 2013, 29(4): 717-728. https://doi.org/10.1007/s10114-012-0520-6

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  Let H₁,H₂ be subgroups of a finite group G. Assume that G=U_i=1^m H₂y_iH₁=U_i=1ⁿ H₁g_jH₁ and that y₁=1, g₁=1. Let D_i be the set consisting of right cosets of H₂ contained in H₂y_iH₁ and let d_j(j=1, ..., n) be the set consisting of right cosets contained in H₁g_jH₁. We define the n×m matrix M_z (z=1, ...,m) whose columns and rows are indexed by D_i and d_j respectively and the (d_k,D_l) entry is |D_zgk∩D_l|. Let M=(M₁,...,M_m). Assume that 1H₁^G and 1H₂^G are semisimple permutation modules of a finite group G. In this paper, by using the matrix M, we give some sufficient and necessary conditions such that 1H₁^G is isomorphic to a submodule of 1H₂^G. As an application, we prove Foulkes' conjecture in special cases.
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  λ-Central BMO Estimates for Commutators of Multilinear Maximal Operators

  Zeng Yan SI, Qing Ying XUE

  Acta Mathematica Sinica. 2013, 29(4): 729-742. https://doi.org/10.1007/s10114-013-1753-8

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  The authors establish λ-central BMO estimates for commutators of maximal multilinear Calderón-Zygmund operators T_Πb^* and multilinear fractional operators I_α,b on central Morrey spaces respectively. Similar results still hold for T_b, T_b^* and I_α,b^*.
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  Almost Sure Convergence Theorem and Strong Stability for Weighted Sums of NSD Random Variables

  Yan SHEN, Xue Jun WANG, Wen Zhi YANG, Shu He HU

  Acta Mathematica Sinica. 2013, 29(4): 743-756. https://doi.org/10.1007/s10114-012-1723-6

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  In this paper, Kolmogorov-type inequality for negatively superadditive dependent (NSD) random variables is established. By using this inequality, we obtain the almost sure convergence for NSD sequences, which extends the corresponding results for independent sequences and negatively associated (NA) sequences. In addition, the strong stability for weighted sums of NSD random variables is studied.
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  The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation

  Shao Yong LAI, Nan LI, Jian ZHANG

  Acta Mathematica Sinica. 2013, 29(4): 757-776. https://doi.org/10.1007/s10114-013-1419-6

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  A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space H^s(R) with s>3/2 is established via a limiting procedure. Provided that the initial momentum (1-∂_x²)u₀ satisfies the sign condition, u₀∈H^s(R)(s>3/2) and u₀∈L¹(R), the existence and uniqueness of global solutions for the equation are shown to be true in the space C([0,∞);H^s(R))∩C¹([0,∞);H^s-1(R)).
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  Convergence in Law to Operator Fractional Brownian Motion of Riemann-Liouville Type

  Hong Shuai DAI

  Acta Mathematica Sinica. 2013, 29(4): 777-788. https://doi.org/10.1007/s10114-012-1180-2

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  In this paper, we extend the well-studied fractional Brownian motion of Riemann-Liouville type to the multivariate case, and the corresponding processes are called operator fractional Brownian motions of Riemann-Liouville type. We also provide two results on approximation to operator fractional Brownian motions of Riemann-Liouville type. The first approximation is based on a Poisson process, and the second one is based on a sequence of I.I.D. random variables.
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  Similarity of Multiplication Operators on the Sobolev Disk Algebra

  Kui JI, Rui SHI

  Acta Mathematica Sinica. 2013, 29(4): 789-800. https://doi.org/10.1007/s10114-013-1745-8

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  In this paper, we give a similarity classification for the multiplication operator M_g on the Sobolev disk algebra R(D) with g analytic on the closure of the unit disk D.
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  Anisotropic Curl-free Wavelet Bases on the Unit Cube

  Ying Chun JIANG

  Acta Mathematica Sinica. 2013, 29(4): 801-814. https://doi.org/10.1007/s10114-012-1604-z

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  This paper deals with the construction of anisotropic curl-free wavelets on the cube [0, 1]³, which satisfies the specific boundary conditions. First, one constructs curl-free wavelets on the unit cube based on one dimensional wavelets on the interval [0, 1] with some boundary conditions. Then, the stability of the corresponding wavelets in curl-free space and the characterization of Sobolev spaces are studied. Finally, one gives a Helmholtz decomposition and the representation of curl and div operators in wavelet coordinates.
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  Dynamically Consistent Nonlinear Evaluations with Their Generating Functions in L^p

  Feng HU

  Acta Mathematica Sinica. 2013, 29(4): 815-832. https://doi.org/10.1007/s10114-013-1715-1

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  In this paper, we study dynamically consistent nonlinear evaluations in L^p (1<p<2). One of our aim is to obtain the following result: under a domination condition, an F_t-consistent evaluation is an ε^g-evaluation in L^p. Furthermore, without the assumption that the generating function g(t, ω, y, z) is continuous with respect to t, we provide some useful characterizations of an ε^g-evaluation by g and give some applications. These results include and extend some existing results.