Principal frequency of p-sub-Laplacians for general vector fields

TL;DR

This paper proves the uniqueness and simplicity of the principal frequency of the p-sub-Laplacian for general vector fields, with applications to specific structures like the Grushin plane and Heisenberg group.

Contribution

It establishes the first eigenvalue properties for p-sub-Laplacians in broad vector field contexts, extending previous results to more general settings.

Findings

Proved uniqueness of the principal eigenvalue.

Established simplicity of the first eigenvalue.

Derived Caccioppoli inequalities for the setting.

Abstract

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.