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\(L^2\) continuity for Fourier integral operators with nonregular phases. (English) Zbl 1079.35107

Summary: We give \(L^2\) continuity results for Fourier integral operators with nonregular phases and amplitudes. More precisely, we consider operators of the type: \[ Av(x)=\int_{\mathbb{R}^n} e^{iS(x, \eta)}a(x,\eta)\widehat v(\eta)d\eta,\;v \in{\mathcal S}(\mathbb{R}^n), \] where \(\widehat v(\eta)=\int e^{-iy\eta}v(y)dy\) is the Fourier transform of \(v\), and \(S,a\) are respectively the phase function and the amplitude of the Fourier integral operator \(A\). The geometry of the phase function is rather standard since it is supposed to be nondegenerate in the sense of Yu. Egorov. The novelty here consists in the regularity conditions we impose on the phase function and the amplitude. Roughly speaking, we assume that the phase function is classical up to second derivatives and of type \((1/2,1/2)\) beyond, in the terminology of Hörmander. The amplitude is also supposed to be of type \((1/2,1/2)\).

MSC:

35S30 Fourier integral operators applied to PDEs
42B25 Maximal functions, Littlewood-Paley theory
46F12 Integral transforms in distribution spaces
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