The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds

Abstract

The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities.

The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

“…One of the first studies goes back to Gaveau [15] who provided an expression for the subelliptic heat kernel on the simplest sub-Riemannian model space, the 3-dimensional Heisenberg group. The subelliptic heat kernel of the 3-dimensional Hopf fibration on S 3 was first studied by Bauer [6] and then, in more details by Baudoin and Bonnefont [3]. The subelliptic heat kernel of the 3-dimensional Hopf fibration on SL(2) was then studied by Bonnefont [10].…”

Hamilton Type Gradient Estimate for the Sub-Elliptic Operators

“…(ii). The type of estimate for the heat kernel (3.2) has been shown in some interesting hypoelliptic models, see [6,7,18,19,23,24]. In Heisenberg(-type) group and three Brownian motion model and generally the nilpotent Lie group of rank two (ρ 1 = 0), Eq.…”

“…It is worth recalling that, H. Q. Li [18] obtained the following optimal gradient estimate for the heat kernel, which plays an important role in the proof of H. Q. Li inequality (see [2,18]) on the Heisenberg (type) groups. See also [6,10,19,23,24] for other various sub-elliptic models.…”

Gradient Estimates for the Heat Semigroup on H-Type Groups

“…4 when p = 1, they have obtained many very interesting applications, including Gross-Poincaré type inequalities, Cheeger type isoperimetric inequalities, and Bobkov type isoperimetric inequalities. For some related works, see for example [7] and [25].…”