Generalizations of the Drift Laplace Equation over the Quaternions in a Class of Grushin-Type Spaces

TL;DR

This paper extends fundamental solutions of the Laplace equation with drift from complex to quaternionic domains in Grushin-type spaces, demonstrating stability as p approaches infinity.

Contribution

It introduces a quaternionic generalization of the drift Laplace equation in Grushin-type spaces, expanding prior complex-based results.

Findings

Solutions are stable under limits as p tends to infinity.

Generalization from complex to quaternionic drift terms.

Extends previous formulas to a broader class of spaces.

Abstract

Beals, Gaveau, and Greiner in 1996 establish a formula for the fundamental solution to the Laplace equation with drift term in Grushin-type planes. The first author and Childers in 2013 expanded these results by invoking a p-Laplace-type generalization that encompasses these formulas while the authors explored a different natural generalization of the p-Laplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity.