Fourier Integral Operators and Gelfand-Shilov Spaces

Wave front set at infinity for tempered ultradistributions and hyperbolic equations

“…In this section, we apply the results proved in [5], [6], [7] and described in the previous section to study the Cauchy problem (1.4). The first main statement concerns well posedness of the Cauchy problem in the spaces S θ θ (R n ), (S θ θ ) (R n ), θ > 1.…”

“…Basic examples are differential operators with polynomial coefficients. A detailed exposition on these symbols can be found in [6], [7], [16], [18], [34]. Here we recall only the basic definitions and properties.…”

“…In the following, we shall call conic neighborhood of the direction ∞ξ o any intersection of an open cone containing ∞ξ o with the complementary set of a closed ball centered in the origin. Following [6], [7] we define two types of cut-off functions.…”

“…In this paper, we study the propagation of singularities for (1.4) using the wave front set at infinity for tempered ultradistributions introduced in [6], [7]. Moreover, we complete the results of [5] about well posedness in S θ θ (R n ), (S θ θ ) (R n ) proving the uniqueness of the solution.…”

“…For a detailed exposition of these results we address the reader to [4], [5], [6], [7]. For a complete description of the properties of Gelfand-Shilov spaces, see [10], [19], [29], [32].…”