• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.499-516
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• 2005

We obtain the type set for the fractional integral operator along the curve $(t,t^2,\;{\alpha}t^3)$ on the three dimensional Heisenberg group when $\alpha\neq{\pm}1/6$. The proof is based on the Fourier inversion formula and the angular Littlewood-Paley decompositions in the Heisenberg group in [5].

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.25-32
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• 1998

Let N be the 3-dimensional Heisenberg group equipped with a left-invariant metric on N. We characterize the Jacobi fields and the conjegate points along a geodesic on N, which points out that Theorem 4 of [1] is not correct.

• Bulletin of the Korean Mathematical Society
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• v.38 no.3
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• pp.517-526
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• 2001

In this paper, we study the continuous wavelet transform on the Heisenberg group H$^n$, and describe the related continuous multiscale analysis. By using the wavelet packet transform we obtain a reconstruction formula on L$^2$(H$^n$).

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.405-417
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• 2008

In this paper we prove some Hardy inequalities in the half space of the Heisenberg group and indicate the sharp constant.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.225-235
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• 2023

We show that metric bisectors with respect to the Korányi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1149-1165
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• 2016

In this paper, we show that there exists no left invariant Riemannian metric h on the Heisenberg group H such that (H, h) is a symmetric Riemannian manifold, and there does not exist any H-invariant metric $\bar{h}$ on the Heisenberg manifold $H/{\Gamma}$ such that the Riemannian connection on ($H/{\Gamma},\bar{h}$) is a Yang-Mills connection. Moreover, we get necessary and sufficient conditions for a group homomorphism of (SU(2), g) with an arbitrarily given left invariant metric g into (H, h) with an arbitrarily given left invariant metric h to be a harmonic and an affine map, and get the totality of harmonic maps of (SU(2), g) into H with a left invariant metric, and then show the fact that any affine map of (SU(2), g) into H, equipped with a properly given left invariant metric on H, does not exist.

• Journal of the Chungcheong Mathematical Society
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• v.16 no.1
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• pp.123-135
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• 2003

We shall study the automorphism group of the quaternionic Heisenberg group $\mathcal{H}_7(\mathbb{H})=\mathbb{R}^3{\tilde{\times}}\mathbb{H}$ which is important to investigate an almost Bieberbach group of a 7-dimensinal infra-nilmanifold and show that Aut$$(\mathbb{R}^3{\tilde{\times}}\mathbb{R}^4){\sim_=}Hom(\mathbb{R}^4,\mathbb{R}^3){\rtimes}O(J;2,2)$$.

• East Asian mathematical journal
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• v.21 no.1
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• pp.113-122
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• 2005

Let $\mathbb{H}^3$ be the 3-dimensional Heisenberg group equipped with a left-invariant metric. In this paper, We characterize the Gaussian curvatures of the geodesic spheres on $\mathbb{H}^3$.

• Kyungpook Mathematical Journal
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• v.45 no.4
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• pp.491-502
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• 2005

The main purpose of this paper is to extend the exponentially convex functions which are defined and exponentially convex on a cylinderical neighborhood in the Heisenberg group. They are expanded in terms of an integral transform associated to the sub-Laplacian operator. Extension of such functions on abelian Lie group are studied in [15].

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1329-1343
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• 2013

In this paper, we define translation surfaces in the 3-dimensional Heisenberg group $\mathcal{H}_3$ obtained as a product of two planar curves lying in planes, which are not orthogonal, and study minimal translation surfaces in $\mathcal{H}_3$.