Affiliations: University of Twente, Department of Applied Mathematics, Drienerlolaan 5, 7522 NB Enschede, The Netherlands E‐mail: [email protected]
Note: [] Work completed while the author was at the Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France.
Abstract: 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<L.
Keywords: Fourier integral operators, pseudodifferential initial value problem
