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

Abstract

We consider a class of hypoelliptic ultraparabolic operators in the form

\[ \L=\sum_{j=1}^m X_j^2+X_0-\partial_t, \]

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 \( \L u = 0 \) on \( \RN \times ]0,T[ \) 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 \( \L \).