Hamilton–Jacobi theory and the heat kernel on Heisenberg groups

By R. Beals, B. Gaveau, P. Greiner. 2000

223 citations recorded by the source corpus.

The subelliptic geometry of Heisenberg groups is worked out in detail and related to complex Hamiltonian mechanics. The two geometric pictures are essential for complete understanding of the heat equation for the subelliptic Laplacian. We give a complete description of the geodesics and obtain precise global estimates and small-time asymptotics of the heat kernel.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/47abbc11bbd2fe08f146c7b7bb04c5202d3dcc32

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Astronomy and Astrophysics

In Statistical and Nonlinear Physics

Citations in the corpus

Coercive Inequalities on Metric Measure Spaces (2009, background): “In [32] it was observed that asymptotics from [27] imply the following precise bound (extending [ 6 ]) on the heat kernel p (at time t = 1) on the three-dimensional Heisenberg group H1:”
Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere (2007, unknown): “ate the error term note ﬁrst the formula Z P Z 1 0 f(Bτσ)dσdµu,v,τ(B) = Z Hn Z 1 0 P(τσ,u,w)P(τ−τσ,w,v)f(w)dσdw which is a consequence of (2.42), and valid for nonnegative measurable f. From (4.7) of [BGG] we have the estimate P(τ,u,0) ≤ Cτ−Q/2e−c|u|2/τ for some constants C,c>0 and therefore, for”
The subelliptic heat kernel on the CR sphere (2011, background): “The motivations for finding explicit formulas are numerous, among them we can cite: sharp constant in functional inequalities (see [6], [14]), computation of the sub-Riemannian metric (see [3]), sharp upper and lower bounds for the heat kernel (see [5],[10]), and semigroup sub-commutations (see [13]”
The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups (2008, methodology): “The idea of using the GFT to compute the hypoelliptic heat kernel is not new: it was already used on the Heisenberg group in [17] at the same time as the Gaveau formula was published in [11], and on all step 2 nilpotent Lie groups in [9, 1].”
The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds (2008, background): “Since, numerous papers have been devoted to the study of this kernel (see for instance [9], [19] and the references therein).”
Precise estimates for the subelliptic heat kernel on H-type groups (2008, methodology): “1) was given in [1] for Heisenberg gr oups of all dimensions.”
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations (2014, background): “Though it does not seem very explicit, this representation of the heat kernel has many applications and can be used to get very sharp estimates and small-time asymptotics (see [20] and [38, 39]).”
Invariant Carnot-Caratheodory metrics on $S^3$, $SO(3)$, $SL(2)$ and lens spaces (2007, unknown): “This theorem and its analogs for SO(3) and SL(2) are useful to give estimates for the fundamental solutions of the hypoelliptic heat equation induced by the sub-Riemannian structure ([5, 12, 17, 19]).”