We study the action of Fourier Integral Operators (FIOs) of Hörmander's type on , . We see, from the Beurling-Helson theorem, that generally FIOs of order zero fail to be bounded on these spaces when , the counterexample being given by any smooth non-linear change of variable. Here we show that FIOs of order are instead bounded. Moreover, this loss of derivatives is proved to be sharp in every dimension , even for phases which are linear in the dual variables. The proofs make use of tools from time-frequency analysis such as the theory of modulation spaces.

RESUMEN

We use the calculus of Fourier integral operators with singular symbols to reconstruct electrical permittivity singularities from remote measurements of scattered electric waves. The reconstruction is microlocal and is uniformly valid in the neighborhood of conically refracted waves.