### Abstract

We prove the global $L^p$-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical H\"ormander classes $S^{m}_{\rho, \delta}(\R^n)$ for parameters $0<\rho\leq 1$, $0\leq \delta<1$. We also consider the regularity of operators with amplitudes in the exotic class $S^{m}_{0, \delta}(\R^n)$, $0\leq \delta< 1$ and the forbidden class $S^{m}_{\rho, 1}(\R^n)$, $0\leq\rho\leq 1.$ Furthermore we show that despite the failure of the $L^2$-boundedness of operators with amplitudes in the forbidden class $S^{0}_{1, 1}(\R^n)$, the operators in question are bounded on Sobolev spaces $H^s(\R^n)$ with $s>0.$ This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

Original language | English |
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Number of pages | 41 |

Journal | Analysis and Mathematical Physics |

Publication status | Submitted - 2020 |

### Keywords

- Fourier integral operators
- Hyperbolic PDEs
- Hörmander classes

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## Cite this

Castilla, A. C., Israelsson, A., & Staubach, W. (2020). Regularity of Fourier integral operators with amplitudes in general Hörmander classes. Manuscript submitted for publication. https://arxiv.org/abs/2003.12878