FLAT TRACE ESTIMATES FOR ANOSOV FLOWS

The paper proves that T(z)=tr^b(e^{-it0 h^{-1} P̃_h(z)} R̃_h(z)) is well-defined and holomorphic on [−C1 h^ε, C1 h^ε]+i[−C2 h,1], with the bound T(z)=O(h^{-2n-2}), via a precise wavefront set analysis for the modified resolvent (Proposition 2.1) and a general flat-trace lemma (Lemma 3.2). The key operator bound is ||A e^{-it0 h^{-1} P̃_h(z)} R̃_h(z) B||_{L^2→L^2}=O(h^{-2}), which then yields the stated O(h^{-2n-2}) flat-trace bound (Theorem 2) . By contrast, the candidate solution hinges on two unsupported steps: (i) it treats e^{-it0 h^{-1} P̃_h(z)} as an h-FIO with canonical relation equal to the flow of p, ignoring the presence of the complex absorbing potential Q (this is exactly why the paper develops Lemma 3.1 instead), and (ii) it asserts ellipticity of P̃_h(z) on WF_h(A_-) when A_- is supported in {|ξ|<1}, which generally includes regions where q=σ_h(Q) may vanish and p=0, so ellipticity fails; thus the proposed parametrix E_h(z) with E_h(z)P̃_h(z)A_-=A_-+O(h^∞) is not justified on the stated support. The paper’s argument avoids these pitfalls by directly controlling WF′_h(R̃_h(z)) and WF′_h(e^{-it0 h^{-1} P̃_h(z)}R̃_h(z)), ensuring disjointness from the diagonal and enabling the flat-trace estimate, whereas the model’s HS-reduction relies on unproven microlocal claims .