Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2023-07-01

    Cup product on relative bounded cohomology

    HeeSook Park

    Abstract : In this paper, we define cup product on relative bounded cohomology, and study its basic properties. Then, by extending it to a more generalized formula, we prove that all cup products of bounded cohomology classes of an amalgamated free product \( G_{1}\ast_{A}G_{2} \) are zero for every positive degree, assuming that free factors \( G_i \) are amenable and amalgamated subgroup \( A \) is normal in both of them. As its consequences, we show that all cup products of bounded cohomology classes of the groups \( \mathbb{Z} \ast \mathbb{Z} \) and \( \mathbb{Z}_{n} \ast_{\mathbb{Z}_{d}}\mathbb{Z}_m \), where \( d \) is the greatest common divisor of \( n \) and \( m \), are zero for every positive degree.

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  • 2023-01-01

    Weak Herz-type Hardy spaces with variable exponents and applications

    Souad Ben Seghier

    Abstract : Let $\alpha\in(0,\infty)$, $p\in(0,\infty)$ and $q(\cdot): {{\mathbb R}}^{n}\rightarrow[1,\infty)$ satisfy the globally log-H\"{o}lder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator $M$ and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley $g$-function and the Littlewood-Paley $g^{\ast}_{\lambda}$-function.

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  • 2023-09-01

    On non-displaceable Lagrangian submanifolds in two-step flag varieties

    Yoosik Kim

    Abstract : We prove that the two-step flag variety $\mathcal{F}\ell(1,n;n+1)$ carries a non-displaceable and non-monotone Lagrangian Gelfand--Zeitlin fiber diffeomorphic to $S^3 \times T^{2n-4}$ and a continuum family of non-displaceable Lagrangian Gelfand--Zeitlin torus fibers when $n > 2$.

  • 2023-01-01

    Two-weighted conditions and characterizations for a class of multilinear fractional new maximal operators

    Rui Li, Shuangping Tao

    Abstract : In this paper, two weight conditions are introduced and the multiple weighted strong and weak characterizations of the multilinear fractional new maximal operator $\mathcal{M}_{\varphi,\beta}$ are established. Meanwhile, we introduce the $S_{(\vec{p},q),\beta}(\varphi)$ and $B_{(\vec{p},q),\beta}(\varphi)$ conditions and obtain the characterization of two-weighted inequalities for $\mathcal{M}_{\varphi,\beta}$. Finally, the relationships of the conditions $S_{(\vec{p},q),\beta}(\varphi)$, $\mathcal{A}_{(\vec{p},q),\beta}(\varphi)$ and $B_{(\vec{p},q),\beta}(\varphi)$ and the characterization of the one-weight $A_{(\vec{p},q),\beta}(\varphi)$ are given.

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  • 2022-11-01

    $(\mathcal{F}, \mathcal{A})$-Gorenstein flat homological dimensions

    V\'ictor Becerril

    Abstract : In this paper we develop the homological properties of the Gorenstein $(\mathcal{L}, \mathcal{A})$-flat $R$-modules $\mathcal{GF}_{(\mathcal{F} (R), \mathcal{A})}$ proposed by Gillespie, \linebreak where the class $\mathcal{A} \subseteq \mathrm{Mod} (R^{\mathrm{op}})$ sometimes corresponds to a duality pair $(\mathcal{L}, \mathcal{A})$. We study the weak global and finitistic dimensions that come with the class $\mathcal{GF}_{(\mathcal{F} (R), \mathcal{A})}$ and show that over a $(\mathcal{L}, \mathcal{A})$-Gorenstein ring, the functor $-\otimes _R-$ is left balanced over $\mathrm{Mod} (R^{\mathrm{op}}) \times \mathrm{Mod} (R)$ by the classes $\mathcal{GF}_{(\mathcal{F} (R^{\mathrm{op}}), \mathcal{A})} \times \mathcal{GF}_{(\mathcal{F} (R), \mathcal{A})}$. When the duality pair is $(\mathcal{F} (R), \mathcal{FP}Inj (R^{\mathrm{op}}))$ we recover the G. Yang's result over a Ding-Chen ring, and we see that is new for $(\mathrm{Lev} (R), \mathrm{AC} (R^{\mathrm{op}}))$ among others.

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  • 2022-11-01

    A new optimal eighth-order family of multiple root finders

    Dejan \'Cebi\'c , Neboj\v sa M. Ralevi\'c

    Abstract : This paper presents a new optimal three-step eighth-order family of iterative methods for finding multiple roots of nonlinear equations. Different from the all existing optimal methods of the eighth-order, the new iterative scheme is constructed using one function and three derivative evaluations per iteration, preserving the efficiency and optimality in the sense of Kung-Traub's conjecture. Theoretical results are verified through several standard numerical test examples. The basins of attraction for several polynomials are also given to illustrate the dynamical behaviour and the obtained results show better stability compared to the recently developed optimal methods.

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  • 2024-01-01

    Gorenstein $FP_{n}$-injective modules with respect to a semidualizing bimodule

    Zhiqiang Cheng, Guoqiang Zhao

    Abstract : Let $S$ and $R$ be rings and $_{S}C_{R}$ a semidualizing bimodule. We introduce the notion of $G_C$-$FP_n$-injective modules, which generalizes $G_C$-$FP$-injective modules and $G_C$-weak injective modules. The homological properties and the stability of $G_C$-$FP_n$-injective modules are investigated. When $S$ is a left $n$-coherent ring, several nice properties and new Foxby equivalences relative to $G_C$-$FP_n$-injective modules are given.

  • 2023-09-01

    Summability in Musielak--Orlicz Hardy spaces

    Jun Liu, Haonan Xia

    Abstract : Let $\varphi: \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be a growth function and $H^{\varphi}(\mathbb{R}^n)$ the Musielak--Orlicz Hardy space defined via the non-tangential grand maximal function. A general summability method, the so-called $\theta$-summability is considered for multi-dimensional Fourier transforms in $H^{\varphi}(\mathbb{R}^n)$. Precisely, with some assumptions on $\theta$, the authors first prove that the maximal operator of the $\theta$-means is bounded from $H^{\varphi}(\mathbb{R}^n)$ to $L^{\varphi}(\mathbb{R}^n)$. As consequences, some norm and almost everywhere convergence results of the $\theta$-means, which generalizes the well-known Lebesgue's theorem, are then obtained. Finally, the corresponding conclusions of some specific summability methods, such as Bochner--Riesz, Weierstrass and Picard--Bessel summations, are also presented.

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  • 2024-01-01

    Complete noncompact submanifolds of manifolds with negative curvature

    Ya Gao , Yanling Gao, Jing Mao , Zhiqi Xie

    Abstract : In this paper, for an $m$-dimensional ($m\geq5$) complete noncompact submanifold $M$ immersed in an $n$-dimensional ($n\geq6$) simply connected Riemannian manifold $N$ with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of $M$, the norm of its weighted mean curvature vector $|\textbf{\emph{H}}_{f}|$ and the weighted real-valued function $f$, we can obtain:$\bullet$ several one-end theorems for $M$; $\bullet$ two Liouville theorems for harmonic maps from $M$ to complete Riemannian manifolds with nonpositive sectional curvature. 

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  • 2022-11-01

    Maximal domains of solutions for analytic quasilinear differential equations of first order

    Chong-Kyu Han, Taejung Kim

    Abstract : We study the real-analytic continuation of local real-analytic solutions to the Cauchy problems of quasi-linear partial differential equations of first order for a scalar function. By making use of the first integrals of the characteristic vector field and the implicit function theorem we determine the maximal domain of the analytic extension of a local solution as a single-valued function. We present some examples including the scalar conservation laws that admit global first integrals so that our method is applicable.

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March, 2024
Vol.61 No.2

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