${L^p}$-Liouville Theorems for Invariant Partial Differential Operators in ${\mathbb{R}^n}$

TL;DR

This paper establishes $L^p$-Liouville theorems for hypoelliptic second-order PDEs that are invariant under Lie group translations in $R^n$, covering solutions and subsolutions.

Contribution

It introduces new $L^p$-Liouville theorems for a class of invariant hypoelliptic PDEs, extending previous results to broader operators and invariance settings.

Findings

Proved $L^p$-Liouville theorems for solutions.

Extended results to subsolutions.

Applicable to hypoelliptic operators invariant under Lie group actions.

Abstract

We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $\mathbb{R}^n$. Results for both solutions and subsolutions are given.