A representation formula for regular functions on the characteristic plane of the second Heisenberg group

TL;DR

This paper develops a representation formula for smooth functions on the characteristic plane of the second Heisenberg group, using a conformal Laplacian and an approximation of the fundamental solution, with stability at characteristic points.

Contribution

It introduces a conformal Laplacian and derives a stable representation formula for functions on the characteristic plane of the second Heisenberg group.

Findings

Established a conformal Laplacian on the characteristic plane.

Derived a representation formula involving gradients and fundamental solutions.

Proved stability of the formula at characteristic points.

Abstract

The aim of this paper is to study a Laplace-type operator and its fundamental solution on the characteristic plane in the Heisenberg group $\mathbb{H}^2$. We introduce a conformal version of the Laplacian and we prove that the distance induced by the immersion in the ambient space is a good approximation of its fundamental solution. We provide in particular a representation formula for smooth functions in terms of the gradient of the function and the gradient of the approximated fundamental solution. This representation formula in the plane is stable up to its characteristic point.