Weighted ${L^p}$-Liouville Theorems for Hypoelliptic Partial Differential Operators on Lie Groups

TL;DR

This paper establishes weighted $L^p$-Liouville theorems for hypoelliptic PDEs on Lie groups, demonstrating uniqueness and providing examples and applications to evolution equations.

Contribution

It introduces new weighted $L^p$-Liouville theorems for hypoelliptic operators on Lie groups, extending classical results with natural right-invariant measures.

Findings

Weighted $L^p$-Liouville theorems for hypoelliptic operators

Application to uniqueness of Cauchy problem for evolution operators

Examples illustrating the applicability of the theorems

Abstract

We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $\mathcal{L}$ on Lie groups $\mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is the right-invariant measure $\check{H}$ of $\mathbb{G}$. We also prove Liouville-type theorems for $C^2$ subsolutions in $L^p(\mathbb{G},\check{H})$. We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator $\mathcal{L}-\partial_t$.