Regularity of Fourier integral operators with amplitudes in general Hörmander classes

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 Hörmander classes $S_{\rho ,\delta }^m({\mathbb{R}}^n)$ for parameters $0<\rho \le 1$, $0\le \delta <1$. We also consider the regularity of operators with amplitudes in the exotic class $S_{0,\delta }^m({\mathbb{R}}^n)$, $0\le \delta <1$ and the forbidden class $S_{\rho ,1}^m({\mathbb{R}}^n)$, $0\le \rho \le 1.$ Furthermore we show that despite the failure of the $L^2$-boundedness of operators with amplitudes in the forbidden class $S_{1,1}^0({\mathbb{R}}^n)$, the operators in question are bounded on Sobolev spaces $H^s({\mathbb{R}}^n)$ with $s>0.$ This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

41 pages

References in corpus (1)

• Global and local regularity of Fourier integral operators on weighted and unweighted spaces