Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-mixing Nonuniformly Hyperbolic Maps

The paper proves (i) an optimal Functional Correlation Bound (FCB) with rate n^{-(β−1)} for systems modelled by two-sided Young towers, and (ii) from any FCB with exponent γ>1, optimal moment bounds for Birkhoff sums and iterated sums. These are stated clearly in Theorems 2.3 and 2.4 and proved via a transfer-operator/one-sided-approximation route plus a weak-dependence scheme in Sections 3–5, including Lemma 3.6 (mixing case), Lemma 4.1 (weak dependence), and the block-decoupling estimates culminating in (5.11) for iterated sums . By contrast, the candidate solution’s Part A (FCB via Young coupling with tail Δ^{-(β−1)}) is plausible and aligns with known coupling heuristics, but Part B contains a critical error: it asserts the inequality |S_{v,w}(n)| ≤ |S_v(n) S_w(n)| + ∑_{i<n} |v_i w_i|, which is false in general (since S_v S_w = S_{v,w}+S_{w,v}+diag, one cannot drop |S_{w,v}|). This invalidates the derivation of the iterated-sum moment bound (Theorem 2.4(b)), whereas the paper proves this bound correctly using a block/independence approximation and moment inequalities . The paper’s arguments are consistent and well grounded in the Young tower framework (Definitions 2.1–2.2; Lemma 3.4; reduction to mixing) .