Estimations l²precisees pour des integrales oscillantes

Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators

“…Now, applying Proposition 3.2 and its analogue concerning the H s,s ,σ ul space (see Proposition 1 of [5]), we obtain the corollary.…”

“…Moreover, applying the same argument and the result of [5] instead of that of [4], one may relax the regularity condition on the phase ϕ and require, for example, only that ∂ α ϕ is in the (global) Hölder space C n+ν+ (R 2n+ν ), > 0, for all multi-indices α such that |α| = 2.…”

“…Unfortunately, we have not been able to produce a proof as simple as that of Theorem 2.1. We follow here the method of proof of Theorem 2 of [5] but not its scheme. Set a (x, y, θ) = a(x, y, θ)ψ( θ) and…”

“…Here, we used essentially Cauchy-Schwarz inequality and the following elementary lemma whose proof is left to the reader (this is Lemma 1 of [5]):…”

Fourier-integral-operator product representation of solutions to first-order symmetrizable hyperbolic systems

“…In the case m = 0, if the symbol σ (z ,z) were of type 1, 0 (or, more generally, of type 0, 0) and the phase function were to be real (l 1 (z, .) = 0), then we could apply the regularity result of [1] (see also [3]). Here, because of the symbol type and the complex phase function we use, we need to prove this regularity result.…”