Geometric Analysis on the Heisenberg Group and Its Generalizations

Fundamental Solutions for a Family of Sub-elliptic PDEs

“…When ∆ X is sub-elliptic, τ g behaves like the square of a distance function, even though it is complex. The following result can be found in [12].…”

“…In general, we have the following result. If (x , t), x = 0, is connected to the origin by an infinite number of geodesics, then the infinity of the number of geodesics connecting (x , t) to (0, 0) is "smaller" than the infinity of the number of geodesics connecting (0, t) to (0, 0); this can be made precise (see [4] and [12]). …”

“…The Heisenberg group and its sub-Laplacian are at the cross-roads of many analysis and geometry domains. A few of these domains are nilpotent Lie groups theory, hypoelliptic second order partial differential equations, strongly pseudoconvex domains in complex analysis, probability theory of degenerate diffusion process, subRiemannian geometry, control theory and semiclassical analysis of quantum mechanics, see e.g., [4], [6], [12], [13], [14], and [19]. Here we give a survey of the behavior of the subRiemannian geodesics induced by a family of sub-elliptic partial differential equations.…”

“…We give special attention to the sub-Laplacian operating on the Heisenberg group and a step 4 subRiemannian manifold, which is the paradigm of the theory. This article is one of a series (see [8], [9], [10], [11], and [12]), whose aim is to study the subRiemannian geometry induced by the sub-Laplacian and its analytic consequences. We also give discuss the link between this theory and diffusion magnetic resonance imaging.…”

Nonlocal heat equations in the Heisenberg group

“…The solution of the equation (1.7) is given by v(z, s, t) = P t * u 0 (z, s), where the convolution product is in the Heisenberg group. He also proves that P t is C ∞ , (see also [1], [10], [11], [18], and [7]). …”