Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups*. Issue 6 (3rd June 2019)

• Record Type: Journal Article
• Authors: Li, Hong-Quan; Zhang, Ye
• Is Part Of: Communications in partial differential equations. Volume 44:Issue 6(2019)
• Page Start: 467
• Page End: 503
• Publication Date: 2019-06-03
• Subjects: Asymptotic behavior -- Grushin operators -- heat kernel -- Heisenberg groups -- precise estimates; 58J37 -- 35B40 -- 35H10 -- 35B45 -- 35K08 -- 43A80 -- 58J35 -- 43A85; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2019.1581802
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three-dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals–Gaveau–Greiner. Third, we prove that the uniform asymptotic behaviour at infinity (so the small-time asymptotic behaviour) of the heat kernel for Grushin operators, obtained by the first author, are still valid in two and three dimensions.