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,τ,ξ) 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,Dt,Dx)+Q(t,x,Dt,Dx) 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 Re(S(t)U,U)L2(Rn) with a lower bound −Ct−1‖〈D〉−1U‖L2(Rn).