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Research Article | Open Access

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
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Abstract

In this paper we study the inversion map and the Kelvin transform on the Heisenberg group 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 H n and determine the best constant of the Hardy-Sobolev type inequality.

Keywords

Heisenberg groupKelvin transformsub-LaplacianHardy-Sobolev type inequalityBest constant
CLC number: 35R03, 35H20, 35A22

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AIMS Mathematics
Volume 10 Issue 8,
August 2025
Pages 19438-19459
DOI: 10.3934/math.2025868

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Mu Z, Zhou F. Kelvin transform on the Heisenberg group revisited and applications to the best constant of Hardy-Sobolev type inequality. AIMS Mathematics, 2025, 10(8): 19438-19459. https://doi.org/10.3934/math.2025868

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Received: 21 July 2025
Revised: 19 August 2025
Accepted: 20 August 2025
Published: 15 August 2025
©2025 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)