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    SEQUENCES OF POWERS OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE
    Yong CHEN, Kei Ji IZUCHI, Kou Hei IZUCHI, Young Joo LEE
    Acta mathematica scientia,Series B    2021, 41 (3): 657-669.   DOI: 10.1007/s10473-021-0301-3
    Abstract72)      PDF       Save
    We consider Toeplitz operators $T_u$ with symbol $u$ on the Bergman space of the unit ball, and then study the convergences and summability for the sequences of powers of Toeplitz operators. We first charactreize analytic symbols $\varphi$ for which the sequence $T^{*k}_\varphi f$ or $T^{k}_\varphi f$ converges to 0 or $\infty$ as $k\to\infty$ in norm for every nonzero Bergman function $f$. Also, we characterize analytic symbols $\varphi$ for which the norm of such a sequence is summable or not summable. We also study the corresponding problems on an infinite direct sum of Bergman spaces as a generalization of our result.
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    CONTINUITY PROPERTIES FOR BORN-JORDAN OPERATORS WITH SYMBOLS IN HÖRMANDER CLASSES AND MODULATION SPACES
    Maurice de GOSSON, Joachim TOFT
    Acta mathematica scientia,Series B    2020, 40 (6): 1603-1626.   DOI: 10.1007/s10473-020-0601-z
    Abstract64)      PDF       Save
    We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hörmander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity, nuclearity and Schatten-von Neumann properties to the Born-Jordan calculus.
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    MARTINGALE INEQUALITIES UNDER G-EXPECTATION AND THEIR APPLICATIONS
    Hanwu LI
    Acta mathematica scientia,Series B    2021, 41 (2): 349-360.   DOI: 10.1007/s10473-021-0201-6
    Abstract50)      PDF       Save
    In this paper, we study the martingale inequalities under $G$-expectation and their applications. To this end, we introduce a new kind of random time, called $G$-stopping time, and then investigate the properties of a $G$-martingale (supermartingale) such as the optional sampling theorem and upcrossing inequalities. With the help of these properties, we can show the martingale convergence property under $G$-expectation.
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    CONTINUOUS DEPENDENCE ON DATA UNDER THE LIPSCHITZ METRIC FOR THE ROTATION-CAMASSA-HOLM EQUATION
    Xinyu TU, Chunlai MU, Shuyan QIU
    Acta mathematica scientia,Series B    2021, 41 (1): 1-18.   DOI: 10.1007/s10473-021-0101-9
    Abstract43)      PDF       Save
    In this article, we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation. Based on defining a Finsler-type norm on the tangent space for solutions, we first establish the Lipschitz metric for smooth solutions, then by proving the generic regularity result, we extend this metric to general weak solutions.
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    GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS
    Yabo REN, Boling GUO, Shu WANG
    Acta mathematica scientia,Series B    2021, 41 (3): 679-702.   DOI: 10.1007/s10473-021-0303-1
    Abstract41)      PDF       Save
    Global in time weak solutions to the $\alpha$-model regularization for the three dimensional Euler-Poisson equations are considered in this paper. We prove the existence of global weak solutions to $\alpha$-model regularization for the three dimension compressible Euler-Poisson equations by using the Fadeo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies $\gamma>\frac{4}{3}$.
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    ASYMPTOTIC STABILITY OF A BOUNDARY LAYER AND RAREFACTION WAVE FOR THE OUTFLOW PROBLEM OF THE HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY
    Lili FAN, Meichen HOU
    Acta mathematica scientia,Series B    2020, 40 (6): 1627-1652.   DOI: 10.1007/s10473-020-0602-y
    Abstract37)      PDF       Save
    This article is devoted to studying the initial-boundary value problem for an ideal polytropic model of non-viscous and compressible gas. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and give a rigorous proof of the asymptotic stability of both the degenerate boundary layer and its superposition with the 3-rarefaction wave under some smallness conditions. New weighted energy estimates are introduced, and the trace of the density and velocity on the boundary are handled by some subtle analysis. The decay properties of the boundary layer and the smooth rarefaction wave also play an important role.
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    A REMARK ON GENERAL COMPLEX (α,β) METRICS
    Hongchuan XIA, Chunping ZHONG
    Acta mathematica scientia,Series B    2021, 41 (3): 670-678.   DOI: 10.1007/s10473-021-0302-2
    Abstract37)      PDF       Save
    In this paper, we give a characterization for the general complex (α,β) metrics to be strongly convex. As an application, we show that the well-known complex Randers metrics are strongly convex complex Finsler metrics, whereas the complex Kropina metrics are only strongly pseudoconvex.
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    ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS
    Xuan Truong LE, Thanh Nhan NGUYEN, Ngoc Trong NGUYEN
    Acta mathematica scientia,Series B    2020, 40 (5): 1171-1184.   DOI: 10.1007/s10473-020-0501-2
    Abstract35)      PDF       Save
    In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator ?=-△+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space.
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    ANALYSIS OF THE GENOMIC DISTANCE BETWEEN BAT CORONAVIRUS RATG13 AND SARS-COV-2 REVEALS MULTIPLE ORIGINS OF COVID-19
    Shaojun PEI, Stephen S. -T. YAU
    Acta mathematica scientia,Series B    2021, 41 (3): 1017-1022.   DOI: 10.1007/s10473-021-0323-x
    Abstract27)      PDF       Save
    The severe acute respiratory syndrome COVID-19 was discovered on December 31, 2019 in China. Subsequently, many COVID-19 cases were reported in many other countries. However, some positive COVID-19 samples had been reported earlier than those officially accepted by health authorities in other countries, such as France and Italy. Thus, it is of great importance to determine the place where SARS-CoV-2 was first transmitted to human. To this end, we analyze genomes of SARS-CoV-2 using k-mer natural vector method and compare the similarities of global SARS-CoV-2 genomes by a new natural metric. Because it is commonly accepted that SARS-CoV-2 is originated from bat coronavirus RaTG13, we only need to determine which SARS-CoV-2 genome sequence has the closest distance to bat coronavirus RaTG13 under our natural metric. From our analysis, SARS-CoV-2 most likely has already existed in other countries such as France, India, Netherland, England and United States before the outbreak at Wuhan, China.
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    EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL (p, q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT
    Fanfan CHEN, Yang YANG
    Acta mathematica scientia,Series B    2020, 40 (6): 1666-1678.   DOI: 10.1007/s10473-020-0604-9
    Abstract26)      PDF       Save
    In this article, we study the following fractional $(p,q)$-Laplacian equations involving the critical Sobolev exponent: \[ (P_{\mu, \lambda}) \begin{cases} (-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u=\mu |u|^{q-2}u +\lambda|u|^{p-2}u + |u|^{p_{s_{1}}^{*}-2}u, & \text{in $\Omega$,} \\ u=0, & \text{in $\mathbb{R}^{N} \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^{N}$ is a smooth and bounded domain, $\lambda,\ \mu >0, \ 0 < s_{2} < s_{1} < 1,\ 1 < q < p < \frac{N}{s_{1}} $. We establish the existence of a non-negative nontrivial weak solution to $(P_{\mu, \lambda})$ by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.
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    GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM
    Qingyou SUN, Yunguang LU, Christian KLINGENBERG
    Acta mathematica scientia,Series B    2020, 40 (5): 1185-1194.   DOI: 10.1007/s10473-020-0502-1
    Abstract24)      PDF       Save
    In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable $s$, the system about the variables $\rho$ and $u$ is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function $P( \rho,s)= {\rm e}^{s} {\rm e}^{-\frac{1}{\rho }}$, which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of (1.1) and (1.2) by adding the artificial viscosity to the Riemann invariants system (2.1). When the amplitude of the first two Riemann invariants $(w_{1}(x,0),w_{2}(x,0))$ of system (1.1) is small, $(w_{1}(x,0),w_{2}(x,0))$ are nondecreasing and the third Riemann invariant $s(x,0)$ is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1].
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    ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES
    Qinghua XU, Yuanping LAI
    Acta mathematica scientia,Series B    2020, 40 (6): 1653-1665.   DOI: 10.1007/s10473-020-0603-x
    Abstract24)      PDF       Save
    Let $\mathcal{K}$ be the familiar class of normalized convex functions in the unit disk. In [14], Keogh and Merkes proved that for a function $f(z)=z+\sum\limits_{k=2}^\infty a_kz^k$ in the class $\mathcal{K}$, \begin{align*} |a_3-\lambda a_2^2|\leq \max \left\{\frac{1}{3}, |\lambda-1|\right\},\ \ \lambda \in \mathbb{C}. \end{align*} The above estimate is sharp for each $\lambda$.
    In this article, we establish the corresponding inequality for a normalized convex function $f$ on $\mathbb{U}$ such that $z=0$ is a zero of order $k+1$ of $f(z)-z$, and then we extend this result to higher dimensions. These results generalize some known results.
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    GENERALIZED ROPER-SUFFRIDGE OPERATOR FOR $\epsilon$ STARLIKE AND BOUNDARY STARLIKE MAPPINGS
    Jie WANG, Jianfei WANG
    Acta mathematica scientia,Series B    2020, 40 (6): 1753-1764.   DOI: 10.1007/s10473-020-0610-y
    Abstract21)      PDF       Save
    This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves $\epsilon$ starlikeness on the open unit ball of a complex Banach space $\mathbb{C}\times X$, where $X$ is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings of order $\alpha$ on the unit ball $B^n$ of ${\mathbb{C}}^{n}$, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order $\alpha$ on $B^n$. Finally, we propose some problems.
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    RETRACTION NOTE: “MINIMAL PERIOD SYMMETRIC SOLUTIONS FOR SOME HAMILTONIAN SYSTEMS VIA THE NEHARI MANIFOLD METHOD”
    Editorial Office of Acta Mathematica Scientia
    Acta mathematica scientia,Series B    2020, 40 (5): 1602-1602.   DOI: 10.1007/s10473-020-0524-8
    Abstract21)      PDF       Save
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    WEAK SOLUTION TO THE INCOMPRESSIBLE VISCOUS FLUID AND A THERMOELASTIC PLATE INTERACTION PROBLEM IN 3D
    Srđan TRIFUNOVIĆ, Yaguang WANG
    Acta mathematica scientia,Series B    2021, 41 (1): 19-38.   DOI: 10.1007/s10473-021-0102-8
    Abstract21)      PDF       Save
    In this paper we deal with a nonlinear interaction problem between an incompressible viscous fluid and a nonlinear thermoelastic plate. The nonlinearity in the plate equation corresponds to nonlinear elastic force in various physically relevant semilinear and quasilinear plate models. We prove the existence of a weak solution for this problem by constructing a hybrid approximation scheme that, via operator splitting, decouples the system into two sub-problems, one piece-wise stationary for the fluid and one time-continuous and in a finite basis for the structure. To prove the convergence of the approximate quasilinear elastic force, we develop a compensated compactness method that relies on the maximal monotonicity property of this nonlinear function.
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    REDUCIBILITY FOR A CLASS OF ANALYTIC MULTIPLIERS ON SOBOLEV DISK ALGEBRA
    Yong CHEN, Ya LIU, Chuntao QIN
    Acta mathematica scientia,Series B    2021, 41 (2): 361-370.   DOI: 10.1007/s10473-021-0202-5
    Abstract19)      PDF       Save
    We prove the reducibility of analytic multipliers $M_\phi$ with a class of finite Blaschke products symbol $\phi$ on the Sobolev disk algebra $R(\mathbb{D})$. We also describe their nontrivial minimal reducing subspaces.
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    ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS
    Junqiang ZHANG, Dachun YANG
    Acta mathematica scientia,Series B    2021, 41 (1): 39-66.   DOI: 10.1007/s10473-021-0103-7
    Abstract19)      PDF       Save
    Let $L:=-\Delta+V$ be the Schrödinger operator on $\mathbb{R}^n$ with $n\geq3$, where $V$ is a non-negative potential satisfying $\Delta^{-1}(V)\in L^\infty(\mathbb{R}^n)$. Let $w$ be an $L$-harmonic function, determined by $V$, satisfying that there exists a positive constant $\delta$ such that, for any $x\in\mathbb{R}^n$, $0<\delta\leq w(x)\leq 1$. Assume that $p(\cdot):\ \mathbb{R}^n\to (0,\,1]$ is a variable exponent satisfying the globally $\log$-Hölder continuous condition. In this article, the authors show that the mappings $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto wf\in H^{p(\cdot)}(\mathbb{R}^n)$ and $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto (-\Delta)^{1/2}L^{-1/2}(f)\in H^{p(\cdot)}(\mathbb{R}^n)$ are isomorphisms between the variable Hardy spaces $H_L^{p(\cdot)}(\mathbb{R}^n)$, associated with $L$, and the variable Hardy spaces $H^{p(\cdot)}(\mathbb{R}^n)$.
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    EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS
    Yaghoub JALILIAN
    Acta mathematica scientia,Series B    2020, 40 (6): 1831-1848.   DOI: 10.1007/s10473-020-0614-7
    Abstract18)      PDF       Save
    In this paper, we study the coupled system of Kirchhoff type equations \begin{equation*} \left\{ \begin{array}{ll} -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla u|^{2}{\rm d}x}\bigg)\Delta u+ u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, & x\in \mathbb{R}^3, \\[3mm] -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla v|^{2}{\rm d}x}\bigg)\Delta v+ v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in \mathbb{R}^3, \\[2mm] u,v\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation*} where $a,b > 0$, $ \alpha, \beta > 1$ and $3 < \alpha+\beta < 6$. We prove the existence of a ground state solution for the above problem in which the nonlinearity is not 4-superlinear at infinity. Also, using a discreetness property of Palais-Smale sequences and the Krasnoselkii genus method, we obtain the existence of infinitely many geometrically distinct solutions in the case when $ \alpha, \beta \geq 2$ and $4\leq\alpha+\beta < 6$.
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    THE PROXIMAL RELATION, REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS
    Yuan LIAN, Xiaojun HUANG, Zhiqiang LI
    Acta mathematica scientia,Series B    2021, 41 (3): 729-752.   DOI: 10.1007/s10473-021-0307-x
    Abstract18)      PDF       Save
    In this paper, we study the proximal relation, regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions. A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation, and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action.
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    MULTIPLE SOLUTIONS FOR THE SCHRÖDINGER-POISSON EQUATION WITH A GENERAL NONLINEARITY
    Yongsheng JIANG, Na WEI, Yonghong WU
    Acta mathematica scientia,Series B    2021, 41 (3): 703-711.   DOI: 10.1007/s10473-021-0304-0
    Abstract18)      PDF       Save
    We are concerned with the nonlinear Schrödinger-Poisson equation \begin{equation} \tag{P} \left\{\begin{array}{ll} -\Delta u +(V(x) -\lambda)u+\phi (x) u =f(u), \\ -\Delta\phi = u^2,\ \lim\limits_{|x|\rightarrow +\infty}\phi(x)=0, \ \ \ x\in \mathbb{R}^3, \end{array}\right. \end{equation} where $\lambda$ is a parameter, $V(x)$ is an unbounded potential and $f(u)$ is a general nonlinearity. We prove the existence of a ground state solution and multiple solutions to problem (P).
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