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

### Chiara Cinti

Università di Bologna, Italy

## Abstract

We consider a class of hypoelliptic ultraparabolic operators in the form

under the assumption that the vector fields $X_1,\ldots,X_m$ 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 $u(\cdot,0)=\varphi$, 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 $\L$.

## Cite this article

Chiara Cinti, Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 20 (2009), no. 2, pp. 145–158

DOI 10.4171/RLM/538