A Composition Estimate for Fourier Series with the Supremum in Time Inside the Sum, and a Question of Chruściel

P. T. Chruściel asked (Oberwolfach Reports 18 (2021)) whether the classical composition estimate in the Sobolev space H^s(T²), s > 1, survives when the functions depend on a parameter t and the supremum over t is taken for each Fourier coefficient separately, inside the weighted sum. With N_s(u) = Σℓ∈Z² (1+|ℓ|)²ˢ sup_t |u_ℓ(t)|²: given a smooth function G with G(0) = 0, is there a constant C₂ = C₂(G, s) such that N_s(G∘φ) ≤ C₂ N_s(φ) whenever N_s(φ) ≤ 1? This estimate is the hypothesis of a conditional existence theorem for the Einstein–Maxwell equations for Robinson–Trautman metrics stated there. We show that the answer is yes, in every dimension d ≥ 1 and for every s > d/2, with explicit constants, and for every G ∈ C^k with G(0) = 0 and k > M + 3/2, where M is the exponent in the following bound. The proof rests on a polynomial bound for exponentials: N_s(eiλφ)1/2 ≤ C (1+|λ|)^M uniformly over the real-valued families with N_s(φ) ≤ 1, with M = s + d/2 if s > 1 + d/2 and M arbitrarily close to (s + d/2)/(s − d/2) otherwise. For an explicit family the left side is at least c λs+d/2, whereas the quantity with the supremum outside the sum grows like λ^s on this family; so the bound is sharp for s > 1 + d/2, and the loss λd/2 is the loss of Parseval's identity under the mode-by-mode supremum. Some smoothness of G is necessary: for every k < s + d/2 the estimate fails for some G ∈ C^k with G(0) = 0. The method is known from weighted Fourier algebras (Leblanc) and from modulation spaces (Reich and Sickel), and the necessity argument follows Katznelson and Leblanc; we did not find the estimate for this space in the literature, and the two closest composition theorems that we know, for modulation spaces, do not cover the case which corresponds to it by analogy. The exact growth rate of the exponentials for d/2 < s ≤ 1 + d/2 and the minimal smoothness of G are not determined, and the constants are very large. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-8415343-001 (Chruściel, Quo Vadis, Mathematical GR?, Oberwolfach Rep. 18 (2021), Report No. 40/2021, Section 3.3).