Journal of theKorean Mathematical SocietyJKMS

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

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• 2022-05-01

Boundedness of Calder\'{o}n-Zygmund operators on inhomogeneous product Lipschitz spaces

Shaoyong He, Taotao Zheng

Abstract : In this paper, we study the boundedness of a class of inhomogeneous Journ\'{e}'s product singular integral operators on the inhomogeneous product Lipschitz spaces. The consideration of such inhomogeneous Journ\'{e}'s product singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators. The key idea used here is to develop the Littlewood-Paley theory for the inhomogeneous product spaces which includes the characterization of a special inhomogeneous product Besov space and a density argument for the inhomogeneous product Lipschitz spaces in the weak sense.

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• 2021-11-01

The Ohm-Rush content function III: Completion, globalization, and power-content algebras

Neil Epstein, Jay Shapiro

Abstract : One says that a ring homomorphism $R \rightarrow S$ is \emph{Ohm-Rush} if extension commutes with arbitrary intersection of ideals, or equivalently if for any element $f\in S$, there is a unique smallest ideal of $R$ whose extension to $S$ contains $f$, called the \emph{content} of $f$. For Noetherian local rings, we analyze whether the completion map is Ohm-Rush. We show that the answer is typically yes' in dimension one, but no' in higher dimension, and in any case it coincides with the content map having good algebraic properties. We then analyze the question of when the Ohm-Rush property globalizes in faithfully flat modules and algebras over a 1-dimensional Noetherian domain, culminating both in a positive result and a counterexample. Finally, we introduce a notion that we show is strictly between the Ohm-Rush property and the weak content algebra property.

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• 2021-11-01

Some numerical radius inequalities for semi-Hilbert space operators

Kais Feki

Abstract : Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. Let $\omega_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbert space $\big(\mathcal{H}, {\langle \cdot, \cdot\rangle}_A\big)$, respectively, where ${\langle x, y\rangle}_A := \langle Ax, y\rangle$ for all $x, y\in\mathcal{H}$. In this paper, we show with different techniques from that used by Kittaneh in \cite{FK} that \begin{equation*} \tfrac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A\le \omega_A^2\left(T\right) \le \tfrac{1}{2}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A. \end{equation*} Here $T^{\sharp_A}$ denotes a distinguished $A$-adjoint operator of $T$. Moreover, a considerable improvement of the above inequalities is proved. This allows us to compute the $\mathbb{A}$-numerical radius of the operator matrix $\left(\begin{smallmatrix} I&T\\ 0&-I \end{smallmatrix}\right)$ where $\mathbb{A}= \text{diag}(A,A)$. In addition, several $A$-numerical radius inequalities for semi-Hilbert space operators are also established.

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• 2021-09-01

Inclusion relations and radius problems for a subclass of starlike functions

Prachi Gupta, Sumit Nagpal, Vaithiyanathan Ravichandran

Abstract : By considering the polynomial function $\phi_{car}(z)=1+z+z^2/2,$ we define the class $\Scar$ consisting of normalized analytic functions $f$ such that $zf'/f$ is subordinate to $\phi_{car}$ in the unit disk. The inclusion relations and various radii constants associated with the class $\Scar$ and its connection with several well-known subclasses of starlike functions is established. As an application, the obtained results are applied to derive the properties of the partial sums and convolution.

• 2022-05-01

Erratum to Static and related critical spaces with harmonic curvature and three Ricci eigenvalues'' [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435--1449]

Jongsu Kim

Abstract : In this erratum, we offer a correction to [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435--1449]. Theorem 1 in the original paper has three assertions (i)-(iii), but we add (iv) after having clarified the argument.

• 2021-05-01

Existence of global solutions to some nonlinear equations on locally finite graphs

Yanxun Chang, Xiaoxiao Zhang

Abstract : Let $G=(V,E)$ be a connected locally finite and weighted graph, $\Delta_p$ be the $p$-th graph Laplacian. Consider the $p$-th nonlinear equation $$-\Delta_pu+h|u|^{p-2}u=f(x,u)$$ on $G$, where $p>2$, $h,f$ satisfy certain assumptions. Grigor'yan-Lin-Yang \cite{GLY2} proved the existence of the solution to the above nonlinear equation in a bounded domain $\Omega\subset V$. In this paper, we show that there exists a strictly positive solution on the infinite set $V$ to the above nonlinear equation by modifying some conditions in \cite{GLY2}. To the $m$-order differential operator $\mathcal{L}_{m,p}$, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.

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• 2021-03-01

Comparison of two desingularizations of the moduli space of elliptic stable maps

Hyenho Lho

Abstract : We study the geometry of the moduli space of elliptic stable maps to projective space. The main component of the moduli space of elliptic stable maps is singular. There are two different ways to desingularize this space. One is Vakil-Zinger's desingularization and the other is via the moduli space of logarithmic stable maps. Our main result is a proof of the direct geometric relationship between these two spaces. For degree less than or equal to 3, we prove that the moduli space of logarithmic stable maps can be obtained by blowing up Vakil-Zinger's desingularization.

• 2021-01-01

Monotonicity criterion and functional inequalities for some $q$-special functions

Khaled Mehrez

Abstract : Our aim in this paper is to derive several new monotonicity properties and functional inequalities of some functions involving the $q$-gamma, $q$-digamma and $q$-polygamma functions. More precisely, some classes of functions involving the $q$-gamma function are proved to be logarithmically completely monotonic and a class of functions involving the $q$-digamma function is showed to be completely monotonic. As applications of these, we offer upper and lower bounds for this special functions and new sharp upper and lower bounds for the $q$-analogue harmonic number harmonic are derived. Moreover, a number of two-sided exponential bounding inequalities are given for the $q$-digamma function and two-sided exponential bounding inequalities are then obtained for the $q$-tetragamma function.

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• 2021-11-01

A Liouville theorem of an integral equation of the Chern-Simons-Higgs type

Qinghua Chen, Yayun Li, Mengfan Ma

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• Ring isomorphisms between closed strings via homological mirror symmetry

Sangwook Lee

J. Korean Math. Soc. 2022; 59(2): 421-438
https://doi.org/10.4134/JKMS.j210435

• The interior gradient estimate for a class of mixed Hessian curvature equations

Jundong Zhou

J. Korean Math. Soc. 2022; 59(1): 53-69
https://doi.org/10.4134/JKMS.j200665

• Goldbach-Linnik type problems with unequal powers of primes

Li Zhu

J. Korean Math. Soc. 2022; 59(2): 407-420
https://doi.org/10.4134/JKMS.j210371

• Generalized Killing structure Jacobi operator for real hypersurfaces in complex hyperbolic two-plane Grassmannians

Hyunjin Lee, Young Jin Suh, Changhwa Woo

J. Korean Math. Soc. 2022; 59(2): 255-278
https://doi.org/10.4134/JKMS.j200614

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• Generalized Killing structure Jacobi operator for real hypersurfaces in complex hyperbolic two-plane Grassmannians

Hyunjin Lee, Young Jin Suh, Changhwa Woo

J. Korean Math. Soc. 2022; 59(2): 255-278
https://doi.org/10.4134/JKMS.j200614

• Grothendieck group for sequences

Xuan Yu

J. Korean Math. Soc. 2022; 59(1): 171-192
https://doi.org/10.4134/JKMS.j210271

• Moment estimate and existence for the solution of neutral stochastic functional differential equation

Huabin Chen, Qunjia Wan

J. Korean Math. Soc. 2022; 59(2): 279-298
https://doi.org/10.4134/JKMS.j210111

• Radius constants for functions associated with a limacon domain

Nak Eun Cho, Anbhu Swaminathan, Lateef Ahmad Wani

J. Korean Math. Soc. 2022; 59(2): 353-365
https://doi.org/10.4134/JKMS.j210246
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