Regularity of Fourier integral operators with amplitudes in general Hörmander classes

Alejandro Castro Castilla, Anders Israelsson, Wolfgang Staubach

Research output: Contribution to journalArticle

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 languageEnglish
Number of pages41
JournalAnalysis and Mathematical Physics
Publication statusSubmitted - 2020

Keywords

  • Fourier integral operators
  • Hyperbolic PDEs
  • Hörmander classes

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