# Cauchy problem for effectively hyperbolic operators with triple characteristics

## TL;DR

This paper proves strong hyperbolicity for a class of effectively hyperbolic operators with triple characteristics at t=0, using energy estimates and symmetrizer construction under a specific condition.

## Contribution

It establishes strong hyperbolicity for operators with triple characteristics under condition (E), extending well-posedness results for such operators with arbitrary lower order terms.

## Key findings

- Operators are strongly hyperbolic under condition (E).
- Energy estimates with weight t^{-N} are effective for analysis.
- A Fefferman-Phong type inequality is proved for the symmetrizer.

## 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 $-C t^{-1}\|\langle D \rangle^{-1}U\|_{L^2(\mathbb R^n)}$.