Estimates for rough Fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators (Q387243)

The authors study the boundedness of Fourier integral operators of the form NEWLINE\[NEWLINEFf(x)= (2\pi)^{-n}\int\exp(i\lambda(x, \eta))\,a(x,\eta)\, \widehat f(\eta)\,d\eta,NEWLINE\]NEWLINE where the phase function \(\lambda(x,\eta)\) satisfies the standard non-degeneracy condition, whereas the amplitude \(a(x,\eta)\) is rough, in particular it satisfies NEWLINE\[NEWLINE\| D^\alpha_\eta a(\cdot, \eta)\|_{L^p(\mathbb{R}^n)}\leq c_\alpha\langle n\rangle^{m-\rho|\alpha|}NEWLINE\]NEWLINE for some \(m\in\mathbb{R}\), \(1\leq p\leq\infty\), \(0\leq\rho\leq 1\). Precise results of \(L^q\)-\(L^r\) boundedness are given, allowing applications to the boundedness of multilinear Fourier integral operators.