\(L^p\) boundedness of Fourier integral operators with rough symbols. (English) Zbl 1498.42012
Summary: In this note, we consider Fourier integral operators \(T_{\phi, a}\) with rough symbols which behave in the spatial variable like an \(L^\infty\) function. Assuming certain weak condition on the phase function we get an \(L^\infty \)-boundedness result of \(T_{\phi, a}\), which generalizes an earlier result of C. E. Kenig and W. Staubach [Stud. Math. 183, No. 3, 249–258 (2007; Zbl 1178.35397)] on pseudo-differential operators. We also prove two boundedness results of \(T_{\phi, a}\) on \(L^\infty\) and \(L^p\), which improve some results of D. Dos Santos Ferreira and W. Staubach [Global and local regularity of Fourier integral operators on weighted and unweighted spaces. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1323.35237)] on Fourier integral operators.
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 35S30 | Fourier integral operators applied to PDEs |
| 47G10 | Integral operators |
| 45P05 | Integral operators |
| 47G30 | Pseudodifferential operators |
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