• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.931-940
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• 2007

Holomorphic mean Lipschitz space is defined in the unit ball of $\mathbb{C}^n$. The membership of the space is expressed in terms of the growth of radial derivatives, which reduced to a classical result of Hardy and Littlewood when n = 1. The membership is also expressed in terms of the growth of tangential derivatives when $n{\ge}2$.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.75-81
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• 2002

Lipschitz stability and Lipschitz ø$_{o}$ - equistability of the functional differential equation x'= B(x)f(t, x, $x_{t}$), $x_{to}$ =$\theta$$_{o}$ are discussed. Sufficient conditions are given using the comparison with the corresponding scalar equation.ion.n.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.21-30
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• 2000

Lipschitz Algebras Lip(X, ${\alpha}$) and lip(X, ${\alpha}$) were first studied by D. R. Sherbert in 1964. B. Pavlovic in 1995 shown that in these algebras, the prime ideals containing a given prime ideal form a chain. In this paper, we show that the above property holds in $Lip^n(X,\;{\alpha})$ and $lip^n(X,\;{\alpha})$, the Lipschitz algebras of finite differentiable functions on a perfect compact place set X.

• Communications of the Korean Mathematical Society
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• v.24 no.2
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• pp.187-195
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• 2009

We investigate the boundary values of the holomorphic mean Lipschitz function. In fact, we prove that the admissible limit exists at every boundary point of the unit ball for the holomorphic mean Lipschitz functions under some assumptions on the Lipschitz order. Moreover, we get embedding theorems of holomorphic mean Lipschitz spaces into Hardy spaces or into the Bloch space on the unit ball in $\mathbb{C}_n$.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.959-971
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• 1997

In the paper we introduce Hausdorff measures which are suitable or the study of Lipschitz regularity of M-harmonic function in the unit ball B in $C^n$. For an M-harmonic function h which satisfies certain integrability conditions, we show that there is an open set $\Omega$, whose Hausdorff content is arbitrarily small, such that h is Lipschitz smooth on $B \backslash \Omega$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.745-765
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• 2021

Let Ω be a proper open subset of ℝⁿ and p(·) : Ω → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the author introduces the "geometrical" variable Hardy spaces H^p(·)_r (Ω) and H^p(·)_z (Ω) on Ω, and then obtains the grand maximal function characterizations of H^p(·)_r (Ω) and H^p(·)_z (Ω) when Ω is a strongly Lipschitz domain of ℝⁿ. Moreover, the author further introduces the "geometrical" variable local Hardy spaces h^p(·)_r (Ω), and then establishes the atomic characterization of h^p(·)_r (Ω) when Ω is a bounded Lipschitz domain of ℝⁿ.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.47-52
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• 2000

We determine the Shilov and Choquet boundaries and the set of peak points of Lipschitz algebras $Lip(X,\;{\alpha})$ for $0<{\alpha}{\leq}1$, and $lip(X,\;{\alpha})$ for $0<{\alpha}<1$, on a compact metric space X. Then, when X is a compact subset of $\mathbb{C}^n$, we define some subalgebras of these Lipschitz algebras and characterize their Shilov and Choquet boundaries. Moreover, for compact plane sets X, we determine the Shilove boundary of them. We also determine the set of peak points of these subalgebras on certain compact subsets X of $\mathbb{C}^n$.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1371-1385
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• 2022

For setting a general weight function on n dimensional complex space ℂⁿ, we expand the classical Fock space. We define Fock type space $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$ of entire functions with a mixed norm, where 0 < p, q < ∞ and t ∈ ℝ and prove that the mixed norm of an entire function is equivalent to the mixed norm of its radial derivative on $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$. As a result of this application, the space $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$ is especially characterized by a Lipschitz type condition.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1333-1354
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• 2019

In authors' paper in 2007, it was shown that the BSE-extension of $C^1_0(R)$, the algebra of continuously differentiable functions f on the real number space R such that f and df /dx vanish at infinity, is the Lipschitz algebra $Lip_1(R)$. This paper extends this result to the case of $C^n_0(R^d)$ and $C^{n-1,1}_b(R^d)$, where n and d represent arbitrary natural numbers. Here $C^n_0(R^d)$ is the space of all n-times continuously differentiable functions f on $R^d$ whose k-times derivatives are vanishing at infinity for k = 0, ${\cdots}$, n, and $C^{n-1,1}_b(R^d)$ is the space of all (n - 1)-times continuously differentiable functions on $R^d$ whose k-times derivatives are bounded for k = 0, ${\cdots}$, n - 1, and (n - 1)-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(R^d)$ has a predual.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.1-17
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• 2004

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton-Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton-Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.