• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.33-69
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• 2023

Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝⁿ → [1, ∞) satisfy the globally log-Hölder 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 $g^*_{\lambda}$-function.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.487-496
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• 2021

In this paper we show that if A ∈ L(X) and B ∈ L(Y), X and Y complex Banach spaces, then A ⊕ B ∈ L(X ⊕ Y) is subscalar if and only if both A and B are subscalar. We also prove that if A, Q ∈ L(X) satisfies AQ = QA and Q^p = 0 for some nonnegative integer p, then A has property (C) (resp. property (𝛽)) if and only if so does A + Q (resp. property (𝛽)). Finally, we show that A ∈ L(X, Y) and B, C ∈ L(Y, X) satisfying operator equation ABA = ACA and BA ∈ L(X) is subscalar with property (𝛿) then both Lat(BA) and Lat(AC) are non-trivial.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.137-152
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• 2014

In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.647-662
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• 2016

In this paper, we study Lipschitz continuous, the boundedness and compactness of the composition operator $C_{\phi}$ acting between the general hyperbolic Bloch type-classes ${\mathcal{B}}^{\ast}_{p,{\log},{\alpha}}$ and general hyperbolic Besov-type classes $F^{\ast}_{p,{\log}}(p,q,s)$. Moreover, these classes are shown to be complete metric spaces with respect to the corresponding metrics.

• Korean Journal of Mathematics
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• v.9 no.2
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• pp.83-98
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• 2001

We characterize ($r,w$)-nuclear operators acting from a Banach space whose dual has weak type $q$ into a Banach space of weak type $p$ by the asymptotic behaviour of their entropy numbers.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.349-361
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• 2022

In this paper, we study the following nonlinear problem: $$\{-\Delta_{p}^{3}(x)u\;=\;{\lambda}V_{1}(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u\;=\;{\Delta}u\;{\Delta}^{2}u\;=\;0\;on\;{\partial}\Omega, $$ under adequate conditions on the exponent functions p, q and the weight function V₁. We prove the existence and nonexistence of eigenvalues for p(x)-triharmonic problem with Navier boundary value conditions on a bounded domain in ℝ^N. Our technique is based on variational approaches and the theory of variable exponent Lebesgue spaces.

• Journal of the Korean Mathematical Society
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• v.42 no.4
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• pp.621-634
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• 2005

Let$P{\rightarrow}M$ be an oriented $S^2-fiber$ bundle over a closed manifold M and let Q be its associated SO(3)-bundle, then we investigate the ring structure of the cohomology of the total space P by constructing the coupling form TA induced from an SO(3) connection A. We show that the cohomology ring of total space splits into those of the base space and the fiber space if and only if the Pontrjangin class $p_1(Q)\;{\in}\;H^4(M;\mathbb{Z})$ vanishes. We apply this result to the twistor spaces of 4-manifolds.

• Honam Mathematical Journal
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• v.6 no.1
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• pp.1-12
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• 1984

Let(X, $\Im$) be a probabilistic metric space with a t-norm. Common fixed point theorems and convergence theorems generalizing the results of Ćirić, Fisher, Sehgal, Istrătescu-Săcuiu and others are proved for three mappings P,S,T on X satisfying $F_{Pu, Pv}(qx){\geq}min\left{F_{Su,Tv}(x),F_{Pu,Su}(x),F_{Pv,Tv}(x),F_{Pu,Tv}(2x),F_{Pv,Su}(2x)\right}$ for every $u, v {\in}X$, all x>0 and some $q{\in}(0, 1)$. One of the main results is extended to uniform spaces. Mathematics Subject Classification (1980): 54H25.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1643-1653
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• 2019

This paper shows that if $1<p<{\infty}$, ${\alpha}{\geq}-n-2$, ${\alpha}>-1-{\frac{p}{2}}$ and f is holomorphic on the unit ball ${\mathbb{B}}_n$, then $${\int_{{\mathbb{B}}_n}}{\mid}Rf(z){\mid}^p(1-{\mid}z{\mid}^2)^{p+{\alpha}}dv_{\alpha}(z)<{\infty}$$ if and only if $${\int_{{\mathbb{B}}_n}}{\int_{{\mathbb{B}}_n}}{\frac{{\mid}f(z)-F({\omega}){\mid}^p}{{\mid}1-(z,{\omega}){\mid}^{n+1+s+t-{\alpha}}}}(1-{\mid}{\omega}{\mid}^2)^s(1-{\mid}z{\mid}^2)^tdv(z)dv({\omega})<{\infty}$$ where s, t > -1 with $min(s,t)>{\alpha}$.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.993-1015
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• 2012

In this paper, we study the quantum sl($n$) representation category using the web space. Specially, we extend sl($n$) web space for $n{\geq}4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl($n$). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial ($n=0$) and Jones polynomial ($n=2$) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.