Operator versions of the van der Corput lemma and Fourier integral operators (Q1897752)

The authors announce very interesting results on oscillatory integral operators of the form \[ Tf(x) = \int_\mathbb{R} e^{i \lambda \varphi (x,y)} f(y)dy \] with various finite type conditions on the phase function. Some of the announced results have already appeared [see in particular the paper in Ann. Math., II. Ser. 140, No. 3, 703-722 (1994; Zbl 0833.43004)].