Singular pseudodifferential calculus for wavetrains and pulses

We generalize the analysis of [12] and develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderón-Vaillancourt Theorem. The remainders in the symbolic calculus are bounded operators on L, whose norm is measured with respect to some small parameter. Our main improvement with respect to [12] consists in showing a regularization effect for the remainders. Due to a nonstandard decay in the frequency variables, the regularization takes place in a scale of anisotropic, and singular, Sobolev spaces. Our analysis allows to extend the results of [12] on the existence of highly oscillatory solutions to nonlinear hyperbolic problems by dropping the compact support condition on the data. The results are also used in our companion work [6] to justify nonlinear geometric optics with boundary amplification, which corresponds to a more singular regime than the one considered in [12]. The analysis is carried out with either an additional real or periodic variable in order to cover problems for pulses or wavetrains in geometric optics.