Peter Greiner

Hamilton–Jacobi theory and the heat kernel on Heisenberg groups☆

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.

Wave kernels related to second-order operators

The wave kernel for a class of second-order subelliptic operators is explicitly computed. This class contains degenerate elliptic and hypo-elliptic operators (such as the Heisenberg Laplacian and the Gru šin operator). Three approaches are used to compute the kernels and to determine their behavior near the singular set. The formulas are applied to study propagation of the singularities. The results are expressed in terms of the real values of a complex function extending the Carnot-Caratheodory distance, and the geodesics of the associated sub-Riemannian geometry play a crucial role in the analysis.

On SubRiemannian Geodesics

We consider a subRiemannian geometry induced by a step 3 subelliptic partial differential operator in ℝ3. Our main result is the characterization of a canonical submanifold through the origin, all of whose points are connected to the origin by infinitely many (subRiemannian) geodesics.

Uniforms hypoelliptic Green's functions☆

Abstract We give a formula for the Green function of certain subelliptic operators with variable rank.

ON GEODESICS IN SUBRIEMANNIAN GEOMETRY II

We study a subRiemannian geometry induced by 2 specific vector fields in ℝ3, and obtain the canonical curve whose tangents provide the missing direction.

A HAMILTONIAN APPROACH TO THE HEAT KERNEL OF A SUBLAPLACIAN ON S2n+1

The heat kernel for the Cauchy-Riemann subLaplacian on S(2n+1) is derived in a manner which is completely analogous to the classical derivation of elliptic heat kernels. This suggests that the classical hamiltonian construction of elliptic heat kernels, with appropriate modifications, does yield heat kernels for subelliptic operators.

THE POSITIVITY OF THE HEAT KERNEL ON HEISENBERG GROUP

An elementary proof is given for the positivity of the heat kernel associated with the sub-Laplacian on Heisenberg group. Starting from an integral representation of the heat kernel, the positivity is shown by choosing a simple, explicit path of integration. The positivity of the heat kernel associated with the Grusin operator then follows.

Approximate identities from Laguerre functions and singular integrals on the Heisenberg group

Suitably scaled Laguerre functions are an approximate identity for multiplicative convolution with test functions on the half line. As an application, we derive a precise connection between the Mikhlin-type expansion of a singular integral operator on a Heisenberg groupHn and its natural restriction toHn modulo the center.

On Green’s functions for subelliptic operators

In this article we give two new derivations of the Green’s function, or fundamental solution, for the sublaplacian associated to the hypersurface Im z n +1 = ❘z❘2 k , z ∈ ℂ n .This Green’s function, given by an Euler transform, is expressed in variables which smoothly relate the generic step two points to the higher step (step 2k) points on the Re z n +1 axis.

Intégrales de Fourier quadratiques et calcul symbolique exact sur le groupe d'Heisenberg

Abstract We construct an exact symbolic calculus for invariant operators on the Heisenberg group using a Fourier transform with a quadratic exponent of Leray type.