We consider the transport equation \ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) =
0 in \OOO\times(0,T), where T>0 and \OOO\subset \R^d is a bounded
domain with smooth boundary \ppp\OOO. First, we prove a Carleman estimate for
solutions of finite energy with piecewise continuous weight functions. Then,
under a further condition which guarantees that the orbits of H intersect
\ppp\OOO, we prove an energy estimate which in turn yields an observability
inequality. Our results are motivated by applications to inverse problems.Comment: 18 pages, 3 figure