L2 Boundedness of the Fourier Integral Operator with Inhomogeneous Phase Functions.

In this paper, we investigate the L2 boundedness of the Fourier integral operator Tø,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions. In particular, if the symbol a ∈ L ∞ S ρ m , the phase function ø satisfies some measure conditions and ∥ ∇ ξ k ϕ (⋅ , ξ) ∥ L ∞ ≤ C ∣ ξ ∣ ϵ − k for all k ≥ 2,ξ ≠ 0, and some ϵ > 0, we obtain that Tø,a is bounded on L2 if m < n 2 min { ρ − 1 , − ϵ 2 } . This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators. Moreover, the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schrödinger operator. [ABSTRACT FROM AUTHOR]