• 충청수학회지
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• 제32권3호
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• pp.349-363
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• 2019

Let ${\mathbb{H}}^5$ be the 5-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we calculate the volumes of geodesic balls in ${\mathbb{H}}^5$. Let $B_e(R)$ be the geodesic ball with center e (the identity of ${\mathbb{H}}^5$) and radius R in ${\mathbb{H}}^5$. Then, the volume of $B_e(R)$ is given by $${\hfill{12}}Vol(B_e(R))\\{={\frac{4{\pi}^2}{6!}}{\left(p_1(R)+p_4(R){\sin}\;R+p_5(R){\cos}\;R+p_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^R}{\frac{{\sin}\;t}{t}}dt\right.}\\{\left.{\hfill{65}}{+q_4(R){\sin}(2R)+q_5(R){\cos}(2R)+q_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^{2R}}{\frac{{\sin}\;t}{t}}dt}\right)}$$ where $p_n$ and $q_n$ are polynomials with degree n.

• 대한수학회보
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• 제16권1호
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• pp.1-9
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• 1979

• 충청수학회지
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• 제19권4호
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• pp.437-443
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• 2006

We study free actions of finite groups on the 3-dimensional nilmanifold for Type 1 and classify all such group actions, up to topological conjugacy. This work supplies missing one in [1, Theorem 3.11.].

• 대한수학회보
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• 제49권6호
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• pp.1335-1348
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• 2012

We prove a Hardy inequality related to Carnot-Carath$\acute{e}$odory distance on H-type groups based on a representation formula on such groups.

• 충청수학회지
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• 제20권3호
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• pp.223-230
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• 2007

We study free actions of finite groups on the 3-dimensional nilmanifold and classify all such group actions, up to topological conjugacy. This work generalize Theorem 3.10 of [1].

• 충청수학회지
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• 제18권2호
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• pp.223-232
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• 2005

We study only free actions of finite groups G on the 3-dimensional nilmanifold, up to topological conjugacy which yields an infra-nilmanifold of type 2.

• 대한수학회지
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• 제56권1호
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• pp.285-287
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• 2019

• 대한수학회지
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• 제42권4호
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• pp.795-826
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• 2005

We study free actions of finite abelian groups on 3­dimensional nilmanifolds. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal nilpotent subgroups of almost Bieberbach groups of finite index, up to affine conjugacy. All such actions are completely classified.

• 대한수학회지
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• 제54권5호
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• pp.1411-1440
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• 2017

We study free actions of finite groups on 3-dimensional nil-manifolds with the first homology ${\mathbb{Z}}^2{\oplus}{\mathbb{Z}}_p$. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal nilpotent subgroups of almost Bieberbach groups of finite index, up to affine conjugacy.

• 대한수학회논문집
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• 제10권1호
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• pp.49-55
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• 1995

This paper is the third of a series in which smoothness properties of function in several variables are discussed. The germ of the whole theory was laid in the works by Folland and Stein [4]. On nilpotent Lie groups, they difined analogues of the classical $L^p$ Sobolev or potential spaces in terms of fractional powers of sub-Laplacian, L and extended several basic theorems from the Euclidean theory of differentaiability to these spaces: interpolation properties, boundedness of singular integrals,..., and imbeding theorems. In this paper we study the analogue to the extension theorem for the Folland-Stein spaces. The analogue to Stein's restriction theorem were studied by M. Mekias [5] and Y.M. Kim [6]. First, we have the space of Bessel potentials on the Heisenberg group introduced by Folland [4].