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BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS Boyong Chen, Liangying Jiang Acta mathematica scientia,Series B    2024, 44 (3): 789-809.   DOI: 10.1007/s10473-024-0301-1 Abstract （158）            Save In this article, we investigate the (big) Hankel operator $H_f$ on the Hardy spaces of bounded strongly pseudoconvex domains $\Omega$ in $\mathbb{C}^n$. We observe that $H_f$ is bounded on $H^p(\Omega)$ ($1< p<\infty$) if $f$ belongs to BMO and we obtain some characterizations for $H_f$ on $H^2(\Omega)$ of other pseudoconvex domains. In these arguments, Amar's $L^p$-estimations and Berndtsson's $L^2$-estimations for solutions of the $\bar{\partial}_b$-equation play a crucial role. In addition, we solve Gleason's problem for Hardy spaces $H^p(\Omega)$ ($1\le p\le\infty$) of bounded strongly pseudoconvex domains. Reference | Related Articles | Metrics

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THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL Han Luo Acta mathematica scientia,Series B    2024, 44 (4): 1189-1199.   DOI: 10.1007/s10473-024-0401-y Abstract （97）            Save In this paper, we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations. Under some suitable conditions, we give the gradient estimates of these maps and establish a Liouville type result. Reference | Related Articles | Metrics

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FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY Ang FU, Mingjin LI, Di YANG Acta mathematica scientia,Series B    2024, 44 (2): 405-419.   DOI: 10.1007/s10473-024-0201-4 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （83）      PDF       Save For an arbitrary solution to the Volterra lattice hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method. In this paper, we define a pair of wave functions of the solution and use them to give an expression of the matrix resolvent; based on this we obtain a new formula for the $k$-point functions for the Volterra lattice hierarchy in terms of wave functions. As an application, we give an explicit formula of $k$-point functions for the even GUE (Gaussian Unitary Ensemble) correlators. Reference | Related Articles | Metrics

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SUMS OF DUAL TOEPLITZ PRODUCTS ON THE ORTHOGONAL COMPLEMENTS OF FOCK-SOBOLEV SPACES Yong CHEN, Young Joo LEE Acta mathematica scientia,Series B    2024, 44 (3): 810-822.   DOI: 10.1007/s10473-024-0302-0 Abstract （81）            Save We consider dual Toeplitz operators on the orthogonal complements of the ock-Sobolev spaces of all nonnegative real orders. First, for symbols in a certain class containing all bounded functions, we study the problem of when an operator which is finite sums of the dual Toeplitz products is compact or zero. Next, for bounded symbols, we construct a symbol map and exhibit a short exact sequence associated with the $C^*$-algebra generated by all dual Toeplitz operators with bounded symbols. Reference | Related Articles | Metrics

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MEAN SENSITIVITY AND BANACH MEAN SENSITIVITY FOR LINEAR OPERATORS Quanquan Yao, Peiyong Zhu Acta mathematica scientia,Series B    2024, 44 (4): 1200-1228.   DOI: 10.1007/s10473-024-0402-x Abstract （78）            Save Let $(X,T)$ be a linear dynamical system, where $X$ is a Banach space and $T:X \to X$ is a bounded linear operator. This paper obtains that $(X,T)$ is sensitive (Li-Yorke sensitive, mean sensitive, syndetically mean sensitive, respectively) if and only if $(X,T)$ is Banach mean sensitive (Banach mean Li-Yorke sensitive, thickly multi-mean sensitive, thickly syndetically mean sensitive, respectively). Several examples are provided to distinguish between different notions of mean sensitivity, syndetic mean sensitivity and mean Li-Yorke sensitivity. Reference | Related Articles | Metrics

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THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX* Yinzheng Sun, Aifang Qu, Hairong Yuan Acta mathematica scientia,Series B    2024, 44 (1): 37-77.   DOI: 10.1007/s10473-024-0102-6 Abstract （73）      PDF       Save We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity $x=x(t)$. We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity $x=x(t)$ is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve $x=x(t)+0$, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field. Reference | Related Articles | Metrics

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LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY* Zhongmin Qian, Xingcheng Xu Acta mathematica scientia,Series B    2024, 44 (5): 1609-1638.   DOI: 10.1007/s10473-024-0501-8 Abstract （66）            Save This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering. Reference | Related Articles | Metrics

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CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY* Thomas E. Cecil Acta mathematica scientia,Series B    2024, 44 (1): 1-36.   DOI: 10.1007/s10473-024-0101-7 Abstract （65）      PDF       Save This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of $S^n$ (or ${\bf R}^n$), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties. Reference | Related Articles | Metrics

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THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL Lvqiao LIU, Juan ZENG Acta mathematica scientia,Series B    2024, 44 (2): 455-473.   DOI: 10.1007/s10473-024-0205-0 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （59）      PDF       Save In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary $L^2$ weighted estimates. Reference | Related Articles | Metrics

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SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION Changlin XIANG, Gaofeng ZHENG Acta mathematica scientia,Series B    2024, 44 (2): 420-430.   DOI: 10.1007/s10473-024-0202-3 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （58）      PDF       Save This paper is a continuation of recent work by Guo-Xiang-Zheng[10]. We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation $\begin{equation*} \Delta^{2}u=\Delta(V\nabla u)+{\rm div}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f\qquad\text{in }B^{4},\end{equation*}$ under the smallest regularity assumptions of $V,w,\omega, F$, where $f$ belongs to some Morrey spaces. This work was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the $L^p$ type regularity theory of [10], and generalizes the work of Du, Kang and Wang [4] on a second order problem to our fourth order problems. Reference | Related Articles | Metrics

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ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng Acta mathematica scientia,Series B    2024, 44 (3): 823-838.   DOI: 10.1007/s10473-024-0303-z Abstract （52）            Save Let $X$ and $Y$ be two normed spaces. Let $\mathcal{U}$ be a non-principal ultrafilter on $\mathbb{N}$. Let $g: X\rightarrow Y$ be a standard $\varepsilon$-phase isometry for some $\varepsilon\geq 0$, i.e., $g(0)=0$, and for all $u,v\in X$, $$|\; |\|g(u)+g(v)\|\pm \|g(u)-g(v)\||-|\|u+v\|\pm\|u-v\||\;|\leq\varepsilon.$$ The mapping $g$ is said to be a phase isometry provided that $\varepsilon=0$. In this paper, we show the following universal inequality of $g$: for each $u^*\in w^*$-exp $\|u^*\|B_{X^*}$, there exist a phase function $\sigma_{u^*}: X\rightarrow \{-1,1\}$ and $\varphi$ $\in$ $Y^*$ with $\|\varphi\|= \|u^*\|\equiv \alpha $ satisfying that $$\;\;\;\;\; |\langle u^*,u\rangle-\sigma_{u^*} (u)\langle \varphi, g(u)\rangle |\leq\frac{5}{2}\varepsilon\alpha ,\;\;{\rm for\;all\;}u\in X.$$ In particular, let $X$ be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all $u^*\in X^*$; (2) the constant $\frac{5}{2}$ can be reduced to $\frac{3}{2}$ provided that $Y^\ast$ is strictly convex; (3) the existence of such a $g$ implies the existence of a phase isometry $\Theta:X\rightarrow Y$ such that $\Theta(u)=\lim\limits_{n,\mathcal{U}}\frac{g(nu)}{n}$ provided that $Y^{**}$ has the $w^*$-Kadec-Klee property (for example, $Y$ is both reflexive and locally uniformly convex). Reference | Related Articles | Metrics

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THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS Zihou ZHANG, Jing ZHOU Acta mathematica scientia,Series B    2024, 44 (2): 445-454.   DOI: 10.1007/s10473-024-0204-1 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （51）      PDF       Save In this paper, we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property. We introduce the concepts of the weak$^*$-weak denting point and the weak$^*$-weak$^*$ denting point of a set. These are the generalizations of the weak$^*$ denting point of a set in a dual Banach space. By use of the weak$^*$-weak denting point, we characterize the very smooth space, the point of weak$^*$-weak continuity, and the extreme point of a unit ball in a dual Banach space. Meanwhile, we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces. Moreover, we define the nearly weak dentability in Banach spaces, which is a generalization of near dentability. We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability. We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the $w$-strong proximinality of every closed convex subset of Banach spaces. Reference | Related Articles | Metrics

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GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH $p$-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE Xiaoshan Wang, Zhongqian Wang, Zhe Jia Acta mathematica scientia,Series B    2024, 44 (3): 909-924.   DOI: 10.1007/s10473-024-0308-7 Abstract （49）            Save This paper is concerned with the following attraction-repulsion chemotaxis system with $p$-Laplacian diffusion and logistic source: $$\left\{\begin{array}{ll}u_{t}=\nabla\cdot(|\nabla u|^{p-2}\nabla u)-\chi \nabla\cdot(u \nabla v)+\xi \nabla\cdot(u \nabla w)+f(u),\;\;\;x\in \Omega,\;t>0,\\v_{t}=\triangle v-\beta v+\alpha u^{k_{1}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\0=\triangle w-\delta w+\gamma u^{k_{2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\u(x,0)=u_{0}(x),\;\;v(x,0)=v_{0}(x),\;\;w(x,0)=w_{0}(x), \;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega.\end{array}\right.$$ The system here is under a homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R}^{n}(n\geq2) $, with $ \chi, \xi, \alpha,\beta,\gamma,\delta, k_{1}, k_{2} >0, p\geq 2$. In addition, the function $f$ is smooth and satisfies that $f(s)\leq\kappa-\mu s^{l}$ for all $s\geq0$, with $\kappa\in \mathbb{R}, \mu>0, l>1$. It is shown that (i) if $l>\max\{ 2k_{1}, \frac{2k_{1}n}{2+n}+\frac{1}{p-1}\}$, then system possesses a global bounded weak solution and (ii) if $k_{2}>\max\{2k_{1}-1, \frac{2k_{1}n}{2+n}+\frac{2-p}{p-1}\}$ with $l>2$, then system possesses a global bounded weak solution. Reference | Related Articles | Metrics

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ENERGY CONSERVATION FOR THE WEAK SOLUTIONS TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW Zhong Tan, Xinliang Li, Hui Yang Acta mathematica scientia,Series B    2024, 44 (3): 851-864.   DOI: 10.1007/s10473-024-0305-x Abstract （48）            Save In this paper, we establish some regularity conditions on the density and velocity fields to guarantee the energy conservation of the weak solutions for the three-dimensional compressible nematic liquid crystal flow in the periodic domain. Reference | Related Articles | Metrics

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COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES (RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)* Xiaoman Duan, Zhuangdan Guan Acta mathematica scientia,Series B    2024, 44 (1): 78-102.   DOI: 10.1007/s10473-024-0103-5 Abstract （47）            Save In this article, we study Kähler metrics on a certain line bundle over some compact Kähler manifolds to find complete Kähler metrics with positive holomorphic sectional (or bisectional) curvatures. Thus, we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry. Reference | Related Articles | Metrics

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ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES Jian CHEN Acta mathematica scientia,Series B    2024, 44 (2): 431-444.   DOI: 10.1007/s10473-024-0203-2 Accepted: 16 October 2023 Abstract （47）      PDF       Save In this note, we mainly make use of a method devised by Shaw [15] for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type $\Omega=\widetilde{\Omega} \backslash \overline{\bigcup_{j=1}^{m}\Omega_j}$, where $\widetilde{\Omega}$ and $\{\Omega_j\}_{j=1}^m\Subset\widetilde{\Omega}$ are bounded pseudoconvex domains in $\mathbb{C}^n$ with smooth boundaries, and $\overline{\Omega}_1,\cdots,\overline{\Omega}_m$ are mutually disjoint. The main results can also be quickly obtained by virtue of [5]. Reference | Related Articles | Metrics

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VARIATIONAL ANALYSIS FOR THE MAXIMAL TIME FUNCTION IN NORMED SPACES* Ziyi Zhou, Yi Jiang Acta mathematica scientia,Series B    2024, 44 (5): 1696-1706.   DOI: 10.1007/s10473-024-0503-6 Abstract （45）            Save For a general normed vector space, a special optimal value function called a maximal time function is considered. This covers the farthest distance function as a special case, and has a close relationship with the smallest enclosing ball problem. Some properties of the maximal time function are proven, including the convexity, the lower semicontinuity, and the exact characterizations of its subdifferential formulas. Reference | Related Articles | Metrics

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HEAT KERNEL ON RICCI SHRINKERS (II)* Yu Li, Bing Wang Acta mathematica scientia,Series B    2024, 44 (5): 1639-1695.   DOI: 10.1007/s10473-024-0502-7 Abstract （44）            Save This paper is the sequel to our study of heat kernel on Ricci shrinkers [29]. In this paper, we improve many estimates in [29] and extend the recent progress of Bamler [2]. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of $\mathbb{F}$-convergence holds naturally on any Ricci flows induced by Ricci shrinkers. Reference | Related Articles | Metrics

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THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS $B_{p}^{n}$ OF $\mathbb{C}^{n}$p OF Cn Jianfei WANG, Yanhui ZHANG Acta mathematica scientia,Series B    2024, 44 (3): 839-850.   DOI: 10.1007/s10473-024-0304-y Abstract （44）            Save By introducing the Carathéodory metric, we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit $p$-ball $B_{p}^{n}$ of $\mathbb{C}^n$. Furthermore, the boundary rigidity theorem for holomorphic self-mappings defined on $B_{p}^{n}$ is obtained. These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for $p=2$, and the unit polydisk for $p=\infty$, respectively. Reference | Related Articles | Metrics

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THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS Xianyong Huang, Xunhuan Deng, Qiru Wang Acta mathematica scientia,Series B    2024, 44 (3): 925-946.   DOI: 10.1007/s10473-024-0309-6 Abstract （43）            Save In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the P ötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than $\theta_{4}(t_1,T)$. The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results. Reference | Related Articles | Metrics