Fourier integral operators of infinite order on Gevrey spaces applications to the Cauchy problem for certain hyperbolic operators

Wave front set at infinity for tempered ultradistributions and hyperbolic equations

“…[20], [21] and in the composition formula, cf. [2], [9], [22], [38]. In some particular cases, (2.7) can be relaxed.…”

“…Before studying the propagation of singularities for (1.4), we give a uniqueness result in S θ θ (R n ), (S θ θ ) (R n ) for the solution. Proofs are just sketched since they follow the same arguments used in [5], [9]. To prove Theorem 3.1, we need some preparation.…”

“…Proof As usual, using Theorem 2.1, we will determine the symbols e j,k (t, s; x, η ) starting from their asymptotic expansions ∑ h≥0 e (h) j,k . Arguing as in [9], we want to obtain the functions e …”

“…t,x and by Cattabriga and Zanghirati [9] who proved semi-global well posedness in G θ (Ω ), Ω open set in R n , for p < 1 θ . In [9] the solution of the Cauchy problem has been expressed in terms of Fourier integral operators defined by symbols of infinite order with respect to ξ introduced in [38].…”

“…In [9] the solution of the Cauchy problem has been expressed in terms of Fourier integral operators defined by symbols of infinite order with respect to ξ introduced in [38]. Concerning well posedness in Gevrey classes we quote also [8], [11], [22], [23], [30].…”