Title: Cauchy problem for effectively hyperbolic operators with triple characteristics,
Abstract: We study the Cauchy problem for effectively hyperbolic operators $P$ with
principal symbol $p(t, x,\tau,\xi)$ having triple characteristics on $t = 0$.
Under a condition (E) we show that such operators are strongly hyperbolic, that
is the Cauchy problem is well posed for $p(t, x,D_t, D_x) + Q(t, x, D_t, D_x)$
with arbitrary lower order term $Q$. The proof is based on energy estimates
with weight $t^{-N}$ for a first order pseudo-differential system, where $N$
depends on lower order terms. For our analysis we construct a non-negative
definite symmetrizer $S(t)$ and we prove a version of Fefferman-Phong type
inequality for ${\rm Re}\, (S(t)U, U)_{L^2({\mathbb R}^n)}$ with a lower bound
$-C t^{-1}\|\langle D \rangle^{-1}U\|_{L^2(\mathbb R^n)}$.