Symmetry of Solutions of Integral Equation in the Heisenberg Group

Zhaobing Cui; Wei Shi

doi:doi:10.11648/j.pamj.20261502.11

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Symmetry of Solutions of Integral Equation in the Heisenberg Group

Zhaobing Cui

, Wei Shi^*

Published in Pure and Applied Mathematics Journal (Volume 15, Issue 2)

Received: 11 February 2026 Accepted: 25 February 2026 Published: 18 March 2026

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Abstract

This paper investigates the existence and symmetry properties of solutions to a class of integral equations on the Heisenberg group. Building upon the moving plane method and Hardy-Littlewood-Sobolev type inequalities, we establish symmetry and monotonicity results for positive solutions of the integral equation. This paper extends classical Euclidean results to the Heisenberg group, highlighting profound interactions between geometry and analysis.

Published in	Pure and Applied Mathematics Journal (Volume 15, Issue 2)
DOI	10.11648/j.pamj.20261502.11
Page(s)	11-17
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Integral Equations, Symmetry, Heisenberg Group

References

[1]	G. Lu, N. Lam, J. Bao, Polyharmonic equations with critical exponential growth in the whole space Rn, Discrete and Continuous Dynamical Systems ., 36 (2015), 577–600.
[2]	J. Prajapat, V. Varghese, Symmetry and classification of solutions to an integral equation in the Heisenberg group Hn, Mathematische Annalen ., 392 (2025), 659-700.
[3]	J. Chen, M. Yang, Existence of positive solutions to nonlinear integral equations on the Heisenberg group, Complex Variables and Elliptic Equations., 68 (2021), 544–567.
[4]	J. Tordecilla, E. Aguila, Quasilinear Schrӧdinger equation with exponential growth on the Heisenberg group, Journal of Elliptic and Parabolic Equations., 11 (2025), 1707-1723.
[5]	P. Pucci, L. Temperini, Entire solutions for some critical equations in the Heisenberg group, Opuscula Mathematica., 42 (2022), 279–303.
[6]	T. Wang, J. Yong, Comparison theorems for backward stochastic Volterra integral equations, Mathematics., 125 (2012), 1756–1798.
[7]	D. Li, Y. Wang, Nonexistence of positive solutions for a system of integral equations on and applications, Communications on Pure and Applied Analysis., 12(2013), 2601-2613.
[8]	C. Sun, Z. Wang, Multi-parameter fractional integration on Heisenberg group, arXiv preprint arXiv: 2510.13411 ., 2025.
[9]	A. Ghosh, R. Singh, Weighted estimates for bilinear fractional integral operator on the Heisenberg group, Bulletin des Sciences Mathématiques., 187(2023), 103310.
[10]	C. Sun, Z. Wang, Stein-Weiss inequality revisit on Heisenberg group, arXiv preprint arXiv: 2511.00845., 2025.
[11]	W. Zhang, Q. Zhou, A Liouville theorem for the 2-Hessian equation on the Heisenberg group, arXiv preprint arXiv: 2510.13411., 2025.
[12]	P. Pucci, L. Temperini, Critical equations with Hardy terms in the Heisenberg group, Rendiconti del Circolo Matematico di Palermo Series 2., 71 (2022), 1049–1077.
[13]	Z. Zhao, H. Dong, W. Ying Kernelfree boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains, Computer Methods in Applied Mechanics and Engineering., 414 (2023), 116163.
[14]	S. Mamon, The Wiener measure on the Heisenberg group and parabolic equations, Journal of Mathematical Sciences., 245 (2020), 155–177.
[15]	G. Ren, Y. Shi, Q. Kang, The tangential k-Cauchy-Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group, Applied Mathematics-A Journal of Chinese Universities., 39 (2024), 181–190.
[16]	H. Chiu, Y. Huang, S. Lai, An application of the moving frame method to integral geometry in the Heisenberg group, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications., 13 (2017), 097.
[17]	X. Han, G. Lu, J. Zhu, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities and integral systems on the Heisenberg group, Nonlinear Analysis., 75 (2012), 4296–4314.
[18]	M. Vitturi, J. Wright Multiparameter singular integrals on the Heisenberg group: Uniform estimates, Transactions of the American Mathematical Society., 373 (2020), 5439-5465.

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Cui, Z., Shi, W. (2026). Symmetry of Solutions of Integral Equation in the Heisenberg Group. Pure and Applied Mathematics Journal, 15(2), 11-17. https://doi.org/10.11648/j.pamj.20261502.11

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ACS Style

Cui, Z.; Shi, W. Symmetry of Solutions of Integral Equation in the Heisenberg Group. Pure Appl. Math. J. 2026, 15(2), 11-17. doi: 10.11648/j.pamj.20261502.11

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AMA Style

Cui Z, Shi W. Symmetry of Solutions of Integral Equation in the Heisenberg Group. Pure Appl Math J. 2026;15(2):11-17. doi: 10.11648/j.pamj.20261502.11

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@article{10.11648/j.pamj.20261502.11,
  author = {Zhaobing Cui and Wei Shi},
  title = {Symmetry of Solutions of Integral Equation in the Heisenberg Group
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {2},
  pages = {11-17},
  doi = {10.11648/j.pamj.20261502.11},
  url = {https://doi.org/10.11648/j.pamj.20261502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261502.11},
  abstract = {This paper investigates the existence and symmetry properties of solutions to a class of integral equations on the Heisenberg group. Building upon the moving plane method and Hardy-Littlewood-Sobolev type inequalities, we establish symmetry and monotonicity results for positive solutions of the integral equation. This paper extends classical Euclidean results to the Heisenberg group, highlighting profound interactions between geometry and analysis.
},
 year = {2026}
}

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TY  - JOUR
T1  - Symmetry of Solutions of Integral Equation in the Heisenberg Group

AU  - Zhaobing Cui
AU  - Wei Shi
Y1  - 2026/03/18
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261502.11
DO  - 10.11648/j.pamj.20261502.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 11
EP  - 17
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261502.11
AB  - This paper investigates the existence and symmetry properties of solutions to a class of integral equations on the Heisenberg group. Building upon the moving plane method and Hardy-Littlewood-Sobolev type inequalities, we establish symmetry and monotonicity results for positive solutions of the integral equation. This paper extends classical Euclidean results to the Heisenberg group, highlighting profound interactions between geometry and analysis.

VL  - 15
IS  - 2
ER  -

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Author Information

Zhaobing Cui

School of Mathematics and Physics, Xinjiang Agricultural University, Urumqi, China

Contact Email

http://orcid.org/0009-0001-1486-4008
Wei Shi

School of Mathematics and Physics, Xinjiang Agricultural University, Urumqi, China

Contact Email

http://orcid.org/0009-0006-6034-3138

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APA Style

Cui, Z., Shi, W. (2026). Symmetry of Solutions of Integral Equation in the Heisenberg Group. Pure and Applied Mathematics Journal, 15(2), 11-17. https://doi.org/10.11648/j.pamj.20261502.11

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ACS Style

Cui, Z.; Shi, W. Symmetry of Solutions of Integral Equation in the Heisenberg Group. Pure Appl. Math. J. 2026, 15(2), 11-17. doi: 10.11648/j.pamj.20261502.11

Copy | Download

AMA Style

Cui Z, Shi W. Symmetry of Solutions of Integral Equation in the Heisenberg Group. Pure Appl Math J. 2026;15(2):11-17. doi: 10.11648/j.pamj.20261502.11

Copy | Download

@article{10.11648/j.pamj.20261502.11,
  author = {Zhaobing Cui and Wei Shi},
  title = {Symmetry of Solutions of Integral Equation in the Heisenberg Group
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {2},
  pages = {11-17},
  doi = {10.11648/j.pamj.20261502.11},
  url = {https://doi.org/10.11648/j.pamj.20261502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261502.11},
  abstract = {This paper investigates the existence and symmetry properties of solutions to a class of integral equations on the Heisenberg group. Building upon the moving plane method and Hardy-Littlewood-Sobolev type inequalities, we establish symmetry and monotonicity results for positive solutions of the integral equation. This paper extends classical Euclidean results to the Heisenberg group, highlighting profound interactions between geometry and analysis.
},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Symmetry of Solutions of Integral Equation in the Heisenberg Group

AU  - Zhaobing Cui
AU  - Wei Shi
Y1  - 2026/03/18
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261502.11
DO  - 10.11648/j.pamj.20261502.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 11
EP  - 17
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261502.11
AB  - This paper investigates the existence and symmetry properties of solutions to a class of integral equations on the Heisenberg group. Building upon the moving plane method and Hardy-Littlewood-Sobolev type inequalities, we establish symmetry and monotonicity results for positive solutions of the integral equation. This paper extends classical Euclidean results to the Heisenberg group, highlighting profound interactions between geometry and analysis.

VL  - 15
IS  - 2
ER  -

Copy | Download