Convex functions on the Heisenberg group

Convex Functions on the Heisenberg Group

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.

On the Second Order Derivatives of Convex Functions on the Heisenberg Group

A classical result of Aleksandrov asserts that convex functions in Rn are twice differentiable a.e., and a rst step to prove it is to show that these functions have second order distributional derivatives which are measures, see [4, pp. 239-245]. On the Heisenberg group, and more generally in Carnot groups, several notions of convexity have been introduced and compared in [3] and [7], and Ambro...

Maximum and Comparison Principles for Convex Functions on the Heisenberg Group

The purpose in this paper is to establish pointwise estimates for a class of convex functions on the Heisenberg group. An integral estimate for classical convex functions in terms of the Monge–Ampère operator det D2u was proved by Aleksandrov, see [3, Theorem 1.4.2]. Such estimate is of great importance in the theory of weak solutions for the Monge–Ampère equation, and its proof revolves around...

The Heisenberg group and Hermite functions

The Fourier Transforms of Lipschitz Functions on the Heisenberg Group

We study the order of magnitude of the Fourier transforms of certain Lipschitz functions on the Heisenberg group Hn. We compare our conclusions with some previous results in the field.