Fourier-integral-operator approximation of solutions to first-order hyperbolic pseudodifferential equations I: convergence in Sobolev spaces

摘要

An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolicfirst-order pseudodifferential equation, $\d_z + a(z,x,D_x)$ with $\Re (a) \geq0$, is constructed as the composition of global Fourier integral operators withcomplex phases. An estimate of the operator norm in $L(H^{(s)},H^{(s)})$ ofthese operators is provided which allows to prove a convergence result for theAnsatz to $U(z',z)$ in some Sobolev space as the number of operators in thecomposition goes to $\infty$.