The heat kernel of sub-Laplace operator on nilpotent Lie groups of step two

Abstract

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in Chang et al. [The Laguerre calculus on the nilpotent Lie group of step two. Preprint; 2019. Available from: http://arxiv.org/abs/1901.06513], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [Generalized Hamilton–Jacobi equation and heat kernel on step two nilpotent Lie groups. In: Gustafsson B, Vasil'ev A, editors. Analysis and mathematical physics. Basel: Birkhüser; 2009 (Trends in mathematics)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis.

COMMUNICATED BY:

• Yongzhi Steve

KEYWORDS:

• Laguerre calculus
• heat kernel
• sub-Laplace operator
• nilpotent Lie groups of step two
• power of sub-Laplace operator

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K08
• 33C45
• 22E25
• 32W05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is partially supported by the National Science Foundation (NSF) grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. The second author is partially supported by the National Nature Science Foundation in China (No. 11801523) and the foundation of China Scholarship Council (No. 201708330519). The third author is partially supported by the National Nature Science Foundation in China (No. 11571305).