15 January 2013, Volume 29 Issue 1

    

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  Bilateral Hardy-type Inequalities

  Mu Fa CHEN

  Acta Mathematica Sinica. 2013, 29(1): 1-32. https://doi.org/10.1007/s10114-012-2316-0

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  This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new in the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.
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  Interior Regularity on the Abreu Equation

  Bo Hui CHEN, An-Min LI, Li SHENG

  Acta Mathematica Sinica. 2013, 29(1): 33-38. https://doi.org/10.1007/s10114-012-0217-x

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  In this paper, we prove the interior regularity for the solution to the Abreu equation in any dimension assuming the existence of the C⁰ estimate.
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  Singular Integrals and Weighted Triebel-Lizorkin and Besov Spaces of Arbitrary Number of Parameters

  Guo Zhen LU, Yue Ping ZHU

  Acta Mathematica Sinica. 2013, 29(1): 39-52. https://doi.org/10.1007/s10114-012-1402-7

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  Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderón’s identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe’s covering lemma in multiparameter setting.
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  On Sullivan’s Conjecture on Cycles in 4-free and 5-free Digraphs

  Hao LIANG, Jun Ming XU

  Acta Mathematica Sinica. 2013, 29(1): 53-64. https://doi.org/10.1007/s10114-012-1538-5

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  For a simple digraph G, let β(G) be the size of the smallest subset X  E(G) such that G?X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β(G) ≤ 0.3819γ(G) if G is a 4-free digraph, and β(G) ≤ 0.2679γ(G) if G is a 5-free digraph. These improve the results of Sullivan in 2008.
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  Real Interpolation between Martingale Hardy and BMO Spaces

  Yan Bo REN, Tie Xin GUO

  Acta Mathematica Sinica. 2013, 29(1): 65-74. https://doi.org/10.1007/s10114-012-1310-x

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  In this paper, we consider the real interpolation with a function parameter between martingale Hardy and BMO spaces. An interpolation theorem for martingale Hardy and BMO spaces is formulated. As an application, real interpolation between martingale Lorentz and BMO spaces is given.
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  Characterizing R-duality in Banach Spaces

  Ole CHRISTENSEN, Xiang Chun XIAO, Yu Can ZHU

  Acta Mathematica Sinica. 2013, 29(1): 75-84. https://doi.org/10.1007/s10114-012-1199-4

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  R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces.
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  Diffeomorphisms with C¹-stably Average Shadowing

  Manseob LEE, Xiao WEN

  Acta Mathematica Sinica. 2013, 29(1): 85-92. https://doi.org/10.1007/s10114-012-1162-4

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  Let M be a closed smooth manifold M, and let f:M→M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f. We show that if f has the C¹-stably average shadowing property on Λ, then Λ admits a dominated splitting.
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  Laplace-Beltrami Differentiability of Positive Definite Kernels on the Sphere

  M. H. CASTRO, V. A. MENEGATTO, C. P. OLIVEIRA

  Acta Mathematica Sinica. 2013, 29(1): 93-104. https://doi.org/10.1007/s10114-012-1067-2

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  This contribution gives results on the action of the Laplace-Beltrami derivative on sufficiently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.
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  (Quasi-)Poisson Enveloping Algebras

  Yan Hong YANG, Yuan YAO, Yu YE

  Acta Mathematica Sinica. 2013, 29(1): 105-118. https://doi.org/10.1007/s10114-012-1041-z

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  We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.
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  Decomposition and Approximation of Multivariate Functions on the Cube

  Zhi Hua ZHANG

  Acta Mathematica Sinica. 2013, 29(1): 119-136. https://doi.org/10.1007/s10114-012-1014-2

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  In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]^d is a finite sum of the form Σ_jΦ_jΨ_j, where each Φ_j can be extended to a smooth periodic function, each Ψ_j is an algebraic polynomial, and each Φ_jΨ_j is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]^d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.
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  Sharp Estimates of p-Adic Hardy and Hardy-Littlewood-Pólya Operators

  Zun Wei FU, Qing Yan WU, Shan Zhen LU

  Acta Mathematica Sinica. 2013, 29(1): 137-150. https://doi.org/10.1007/s10114-012-0695-x

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  In this paper we get the sharp estimates of the p-adic Hardy and Hardy-Littlewood-Pólya operators on L^q(|x|_p^α dx). Also, we prove that the commutators generated by the p-adic Hardy operators (Hardy-Littlewood-Pólya operators) and the central BMO functions are bounded on L^q(|x|_p^α dx), more generally, on Herz spaces.
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  A Normal Criterion about Two Families of Meromorphic Functions Concerning Shared Values

  Xiao Jun LIU, San Hua LI, Xue Cheng PANG

  Acta Mathematica Sinica. 2013, 29(1): 151-158. https://doi.org/10.1007/s10114-012-0600-7

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  In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D ⊂ C, a₁, a₂, a₃, a₄ be four distinct finite complex numbers. If G is normal, and for every f ∈ F, there exists g ∈ G such that f(z) and g(z) share the values a₁, a₂, a₃, a₄, then F is normal on D.
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  A Trust Region Affine Scaling Method for Bound Constrained Optimization

  Xiao WANG

  Acta Mathematica Sinica. 2013, 29(1): 159-182. https://doi.org/10.1007/s10114-012-0593-2

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  We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values. Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported.
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  Homogeneous Approximation Property for Wavelet Frames with Matrix Dilations, II

  Zhi Jing ZHAO, Wen Chang SUN

  Acta Mathematica Sinica. 2013, 29(1): 183-192. https://doi.org/10.1007/s10114-012-0546-9

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  The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.
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  The Ergodicity of a Class of Almost Anosov Systems

  Yun Hua ZHOU

  Acta Mathematica Sinica. 2013, 29(1): 193-198. https://doi.org/10.1007/s10114-012-0406-7

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  In this paper, we prove that some volume-preserving almost Anosov systems are ergodic if they are essentially accessible. The key idea is that there are stable and unstable manifolds with uniform size on the orbits of the hyperbolic points for these systems.
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  The Clustering Coefficient and the Diameter of Small-world Networks

  Lei GU, Hui Lin HUANG, Xiao Dong ZHANG

  Acta Mathematica Sinica. 2013, 29(1): 199-208. https://doi.org/10.1007/s10114-012-0387-6

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  The small-world network, proposed by Watts and Strogatz, has been extensively studied for the past over ten years. In this paper, a generalized small-world network is proposed, which extends several small-world network models. Furthermore, some properties of a special type of generalized small-world network with given expectation of edge numbers have been investigated, such as the degree distribution and the isoperimetric number. These results are used to present a lower and an upper bounds for the clustering coefficient and the diameter of the given edge number expectation generalized small-world network, respectively. In other words, we prove mathematically that the given edge number expectation generalized small-world network possesses large clustering coefficient and small diameter.