# 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

$L=j=1∑m X_{j}+X_{0}−∂_{t},$

under the assumption that the vector fields $X_{1},…,X_{m}$ and $X_{0}−∂_{t}$ are invariant with respect to a suitable homogeneous Lie group $G$. We show that if $u,v$ are two solutions of $Lu=0$ on $R_{N}×]0,T[$ and $u(⋅,0)=φ$, then each of the following conditions: $∣u(x,t)−v(x,t)∣$ can be bounded by $Mexp(c∣x∣_{G})$, or both $u$ and $v$ are non negative, implies $u≡v$. 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