# 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

\[ \L=\sum_{j=1}^m X_j^2+X_0-\partial_t, \]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 \( \L u = 0 \) on \( \RN \times ]0,T[ \) 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