Generalizations of a Laplacian-Type Equation in the Heisenberg Group and a Class of Grushin-Type Spaces

TL;DR

This paper extends the fundamental solutions of Laplacian-type equations from specific sub-Riemannian spaces to a broader class, revealing that naive generalizations fail but alternative approaches preserve key properties.

Contribution

It introduces a generalized p-Laplace-type equation in the Heisenberg group and Grushin spaces, demonstrating which generalizations maintain essential properties.

Findings

The straightforward p-Laplace extension does not retain desired properties.

Alternative generalizations preserve natural properties of the original equations.

Provides insights into the structure of sub-Riemannian Laplacian-type equations.

Abstract

Beals, Gaveau and Greiner (1996) find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This solution is related to the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces (Bieske-Gong, 2006) and the Heisenberg group (Capogna, Danielli, Garofalo, 1997). We extend the 2-Laplace-type equation to a p-Laplace-type equation. We show that the obvious generalization does not have desired properties, but rather, our generalization preserves some natural properties.