A new class of Fourier Integral Operators (FIOs, for short) is defined. Phase and amplitude functions are chosen in the so-called SG symbol classes, the former with the additional requirements of being of order (1,1), real-valued and suitably growing at infinity. These FIOs turn out to be continuous on the space of rapidly decreasing functions and of temperate distributions. Results about the composition of SG-FIOs with SG-pseudodifferential operators and about the composition of a SG-FIO with its L^2-adjoint are proved. These allow to obtain results about the existence of parametrices for elliptic FIOs, the continuity on the SG-Sobolev Spaces.
Fourier integral operators in SG classes I: Composition theorems and action on SG Sobolev spaces
CORIASCO, Sandro
1999-01-01
Abstract
A new class of Fourier Integral Operators (FIOs, for short) is defined. Phase and amplitude functions are chosen in the so-called SG symbol classes, the former with the additional requirements of being of order (1,1), real-valued and suitably growing at infinity. These FIOs turn out to be continuous on the space of rapidly decreasing functions and of temperate distributions. Results about the composition of SG-FIOs with SG-pseudodifferential operators and about the composition of a SG-FIO with its L^2-adjoint are proved. These allow to obtain results about the existence of parametrices for elliptic FIOs, the continuity on the SG-Sobolev Spaces.
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