• Title, Summary, Keyword: n-Lipschitz

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HOLOMORPHIC FUNCTIONS SATISFYING MEAN LIPSCHITZ CONDITION IN THE BALL

  • Kwon, Ern-Gun;Koo, Hyung-Woon;Cho, Hong-Rae
    • 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 $n{\ge}2$.

PRIME IDEALS IN LIPSCHITZ ALGEBRAS OF FINITE DIFFERENTIABLE FUNCTIONS

  • EBADIAN, ALI
    • 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.

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FATOU THEOREM AND EMBEDDING THEOREMS FOR THE MEAN LIPSCHITZ FUNCTIONS ON THE UNIT BALL

  • Cho, Hong-Rae;Lee, Jin-Kee
    • 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 $\mathbb{C}_n$.

BOUNDARIES AND PEAK POINTS OF LIPSCHITZ ALGEBRAS

  • MAHYAR, H.
    • 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 $\mathbb{C}^n$.

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BANACH FUNCTION ALGEBRAS OF n-TIMES CONTINUOUSLY DIFFERENTIABLE FUNCTIONS ON Rd VANISHING AT INFINITY AND THEIR BSE-EXTENSIONS

  • Inoue, Jyunji;Takahasi, Sin-Ei
    • 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.

HOLOMORPHIC MEAN LIPSCHITZ FUNCTIONS ON THE UNIT BALL OF ℂn

  • Kwon, Ern Gun;Cho, Hong Rae;Koo, Hyungwoon
    • Journal of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.189-202
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    • 2013
  • On the unit ball of $\mathbb{C}^n$, the space of those holomorphic functions satisfying the mean Lipschitz condition $${\int}_0^1\;{\omega}_p(t,f)^q\frac{dt}{t^1+{\alpha}q}\;<\;{\infty}$$ is characterized by integral growth conditions of the tangential derivatives as well as the radial derivatives, where ${\omega}_p(t,f)$ denotes the $L^p$ modulus of continuity defined in terms of the unitary transformations of $\mathbb{C}^n$.

PROJECTIVE LIMIT OF A SEQUENCE OF BANACH FUNCTION ALGEBRAS AS A FRECHET FUNCTION ALGEBRA

  • Sady. F.
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.259-267
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    • 2002
  • Let X be a hemicompact space with ($K_{n}$) as an admissible exhaustion, and for each n $\in$ N, $A_{n}$ a Banach function algebra on $K_{n}$ with respect to $\parallel.\parallel_n$ such that $A_{n+1}\midK_{n}$$\subsetA_n$ and${\parallel}f{\mid}K_n{\parallel}_n{\leq}{\parallel}f{\parallel}_{n+1}$ for all f$\in$$A_{n+1}$, We consider the subalgebra A = { f $\in$ C(X) : $\forall_n\;{\epsilon}\;\mathbb{N}$ of C(X) as a frechet function algebra and give a result related to its spectrum when each $A_{n}$ is natural. We also show that if X is moreover noncompact, then any closed subalgebra of A cannot be topologized as a regular Frechet Q-algebra. As an application, the Lipschitzalgebra of infinitely differentiable functions is considered.d.

LIPSCHITZ TYPE CHARACTERIZATIONS OF HARMONIC BERGMAN SPACES

  • Nam, Kyesook
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.1277-1288
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    • 2013
  • Wulan and Zhu [16] have characterized the weighted Bergman space in the setting of the unit ball of $C^n$ in terms of Lipschitz type conditions in three different metrics. In this paper, we study characterizations of the harmonic Bergman space on the upper half-space in $R^n$. Furthermore, we extend harmonic analogues in the setting of the unit ball to the full range 0 < p < ${\infty}$. In addition, we provide the application of characterizations to showing the boundedness of a mapping defined by a difference quotient of harmonic function.