Kelvin transform on the Heisenberg group revisited and applications to the best constant of Hardy-Sobolev type inequality

Zimiao Mu; Feng Zhou; Zimiao Mu; Feng Zhou

doi:10.3934/math.2025868

AIMS Mathematics

2025, Volume 10, Issue 8: 19438-19459. doi: 10.3934/math.2025868

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Kelvin transform on the Heisenberg group revisited and applications to the best constant of Hardy-Sobolev type inequality

Zimiao Mu ,
Feng Zhou ^,

School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, China

Received: 21 July 2025 Revised: 19 August 2025 Accepted: 20 August 2025 Published: 26 August 2025
MSC : 35R03, 35H20, 35A22

In this paper we study the inversion map and the Kelvin transform on the Heisenberg group $ \mathbb{H}^{n} $. We first analyze the invariance of the Kelvin transform and provide an algebraic proof to the formula involving the sub-Laplacian. Furthermore, we apply the formula to seek the cylindrically symmetric solution to a sub-elliptic equation on $ \mathbb{H}^{n} $ and determine the best constant of the Hardy-Sobolev type inequality.
- Heisenberg group,
- Kelvin transform,
- sub-Laplacian,
- Hardy-Sobolev type inequality,
- Best constant
Citation: Zimiao Mu, Feng Zhou. Kelvin transform on the Heisenberg group revisited and applications to the best constant of Hardy-Sobolev type inequality[J]. AIMS Mathematics, 2025, 10(8): 19438-19459. doi: 10.3934/math.2025868

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Abstract

In this paper we study the inversion map and the Kelvin transform on the Heisenberg group $ \mathbb{H}^{n} $. We first analyze the invariance of the Kelvin transform and provide an algebraic proof to the formula involving the sub-Laplacian. Furthermore, we apply the formula to seek the cylindrically symmetric solution to a sub-elliptic equation on $ \mathbb{H}^{n} $ and determine the best constant of the Hardy-Sobolev type inequality.

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