FBI Transform in Gevrey Classes and Anosov Flows

Scattering rigidity for analytic metrics

“…We endow T * M with the corresponding real-analytic Kohn-Nirenberg metric g KN = dx 2 + dξ 2 ξ 2 , and consider the associated Grauert tube (T * M) ǫ . This is a conic, complex, pseudo-convex neighbourhood of T * M. We define on it a Japanese bracket (T * M) ǫ ∋ α → |α| ∈ [1, +∞[ and the distance d KN associated to the Kohn-Nirenberg metric (see pages 23-24 in [BJ20] for further discussion of these definitions). For a subset A of M, we will write T * A M for the restriction of T * M to A.…”

“…(4.5) for α, β ∈ (T * M) ǫ ′ . The remaining integral over U only involves real-analytic functions that are exponentially small on the boundary of U , and can consequently be dealt with following exactly the same steps as in the proof of Lemmas 2.9 and 2.10 in [BJ20] in the analytic case pp.110-111 (here we do not need the much more complicated proof required for the Gevrey case). For the convenience of the reader, we recall here the main lines of the argument.…”

“…Before we give the proof of Lemma 4.15, let us link H ω I Re p to Φ t . From equation (2.3) in [BJ20], we see that for a smooth function f , we have in local coordinates x = x+iy, ξ = ξ+iη…”

“…We emphasize that the fact that the operator Π g 0 is an analytic pseudo-differential operator was proved for simple analytic metrics by Stefanov and Uhlmann [SU05]. In our case, the analysis involved is significantly more consequent due to the trapped set and the result can not be deduced from [SU05]; we need to rely on the recent monograph [BJ20]. We can then use an argument in the same spirit as [LTU03] to embed analytically (M, g) into L 2 (N ǫ ) using Π 0 , where N ǫ is an ǫ-collar neighborhood of N := ∂M , and show that the image of the embedding in L 2 (N ǫ ) is entirely determined by the scattering data.…”

“…Using the work of the first and last author [BJ20], we are able to show in Section 4 that, if g is in addition analytic, the Schwartz kernel Π g 0 (x, x ′ ) of Π g 0 is analytic outside the diagonal. This requires in particular to prove radial estimates for the analytic wave front set (Proposition 4.13).…”