Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators

Abstract

We consider a class of hypoelliptic ultraparabolic operators in the form

under the assumption that the vector fields $X_1,\dots ,X_m$ and $X_0-{\partial }_t$ are invariant with respect to a suitable homogeneous Lie group $\mathbb{G}$. We show that if $u,v$ are two solutions of $\mathcal{L}u=0$ on ${\mathbb{R}}^N\times ]0,T[$ and $u(\cdot ,0)=\phi$, then each of the following conditions: $|u(x,t)-v(x,t)|$ can be bounded by $M\exp (c|x{|}_{\mathbb{G}}^2)$, or both $u$ and $v$ are non negative, implies $u\equiv v$. We use a technique which relies on a pointwise estimate of the fundamental solution of $\mathcal{L}$.