Corrigendum to “Center Manifolds without a Phase Space”

The paper defines a translation-equivariant cutoff χ via [χ(u)](x)=∫ χ(‖θ(·−y)u‖_{H^1}) θ(x−y) u(x) dy and proves properties (i)–(v) for χ and their scaled versions for χ_ε(u)=εχ(u/ε), including mapping H^1_{−η}→H^1_{−η}, χ ◦ τ=τ ◦ χ, identity on small H^1_u balls, uniform H^1_u bounds, and a global Lipschitz bound independent of ε (Lemma 2 and Corollary 3) . The candidate solution implements the same operator in the equivalent form χ_ε(u)=A_ε(u)u with A_ε=θ*α_{u,ε} and establishes (i)–(v) by a different route: a W^{1,∞} amplitude/product estimate, a Schur-type integral bound, and a uniform Fréchet-derivative estimate to deduce global Lipschitz continuity. This approach is consistent with the paper’s definition and conclusions, though the technical details differ from the paper’s patchwise H^1 estimates and direct Lipschitz splitting arguments for χ (and scaling to χ_ε) . The only minor point to flag in the paper is that part (i) sketches a bound using H^1_u notation while the domain is H^1_{−η}; this is a presentation choice, not a substantive gap. Overall, both are correct.