Analytic Smoothing and Nekhoroshev Estimates for Hölder Steep Hamiltonians

The uploaded paper proves Theorem 1.1 with exponents a=(ℓ−1)/(2n α1⋯αn−2)+1/2 and b=1/(2n α1⋯αn−1), and stability time T(ε) ≤ C''_T/(|ln ε|^{ℓ−1} ε^a), radius R(ε) ≤ C''_I ε^b, via analytic smoothing with a weighted Fourier estimate (Theorem 4.1) combined with a Pöschel normal form and a steep resonant geometry/trap; see the statement and proof roadmap in the paper’s Theorem 1.1, Lemma B.1, and Section 5 (notably (5.23)) . The candidate solution reproduces the final exponents but its derivation departs materially from the paper: it selects s≈c/|ln ε| (instead of the ε-dependent choice in (5.23)), asserts an extra polynomial-in-K remainder term ||f_s^*|| ≲ K^{-(ℓ−1)}||f_s|| which is not established by the cited normal-form lemma, and misplaces the logarithmic factor when computing T from μ. In the paper, the dominant remainder is ε s^{ℓ−1} (from smoothing), with e^{-Ks} controlled by choosing K,s as in (5.23), yielding μ∼ε^{1+a(ℓ−1)}|ln ε|^{ℓ−1} and T∼1/(|ln ε|^{ℓ−1} ε^{a(ℓ−1)+1/2}) , whereas the candidate alternately claims μ∼ε/|ln ε|^{ℓ−1} and μ∼ε^{1+a(ℓ−1)}/|ln ε|^{ℓ−1}, which would put |ln ε| in the numerator of T. Thus, while the end exponents match, the model’s proof is internally inconsistent and uses an unproven estimate.