Parametrix for a hyperbolic initial value problem with dissipation in some region

We construct parametrices for initial value problems of the form $\[(*)\quad (\curpartial _{z}-\mathrm {i}A(z,x,D_{x})+B(z,x,D_{x}))u=0,\quad z>z_{0},\quad u(z_{0},\cdot)=u_{0},\]$ where $(z,x)\in \mathbb{R} \times \mathbb{R} ^{n}$ , A(z,x,Dx) is a family of order 1 pseudodifferential operators with homogeneous real principal symbol a(z,x,ξ), and B(z,x,Dx) is a family of order γ>0 pseudodifferential operators with non‐negative homogeneous real principal symbol b(z,x,ξ). The parametrix is a family of pseudodifferential operators when A=0, and a Fourier integral operator with real phase function if A≠0. A priori this leads to symbols of type $(\rho,\delta)=(1-\frac{\gamma}{2},\frac{\gamma}{2})$ , which limits our construction to γ<1, and leads to operators with a complicated symbol calculus in the case γ=1. With an additional assumption on B we obtain symbols of type $(\rho,\delta)=(1-\frac{\gamma}{L},\frac{\gamma}{L})$ , for some L≥2. The assumption implies in particular that the first L−1 derivatives of b vanish where b=0. Parametrices for (*) are constructed for the case when 2γ