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

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 $mathcal{L}-partial_t$.

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

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.

On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ partial_t u (x,t) = sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) qquad (x,t) in mathbb{R}^N times, ]- infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.

Hardy Type Inequalities for $Delta_lambda$-Laplacians

We derive Hardy type inequalities for a large class of sub-elliptic operators that belong to the class of $Delta_lambda$-Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin type operators $$ Delta_{x}+ |x|^{2alpha}Delta_{y},qquad (x,y)inmathbb{R}^{N_1}timesmathbb{R}^{N_2}, alphageq 0, $$ which were proved to be sharp.