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 \(H^s(\mathbb{T}^2)\), \(s>1\), 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 \(\mathcal{N}_s(u)\)\({}=\sum_{\ell\in\mathbb{Z}^2}(1\)\({}+\lvert \ell\rvert)^{2s}\sup_t\lvert u_\ell(t)\rvert^2\): given a smooth function \(G\) with \(G(0)=0\), is there a constant \(C_2=C_2(G,s)\) such that \(\mathcal{N}_s(G\circ\varphi)\)\({}\le C_2\,\mathcal{N}_s(\varphi)\) whenever \(\mathcal{N}_s(\varphi)\)\({}\le1\)? 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 \(d\ge1\) and for every \(s>\frac d2\), with explicit constants, and for every \(G\in C^k\) with \(G(0)=0\) and \(k>M+\frac32\), where \(M\) is the exponent in the following bound. The proof rests on a polynomial bound for exponentials: \(\mathcal{N}_s(e^{i\lambda\varphi})^{1/2}\)\({}\le C(1\)\({}+\lvert \lambda\rvert)^{M}\) uniformly over the real-valued families with \(\mathcal{N}_s(\varphi)\)\({}\le1\), with \(M=s+\frac d2\) if \(s>1+\frac d2\) and \(M\) arbitrarily close to \((s\)\({}+\frac d2)\)\(/(s\)\({}-\frac d2)\) otherwise. For an explicit family the left side is at least \(c\,\lambda^{s+d/2}\), whereas the quantity with the supremum outside the sum grows like \(\lambda^{s}\) on this family; so the bound is sharp for \(s>1+\frac d2\), and the loss \(\lambda^{d/2}\) is the loss of Parseval's identity under the mode-by-mode supremum. Some smoothness of \(G\) is necessary: for every \(k<s+\frac d2\) the estimate fails for some \(G\in C^k\) 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 \(\frac d2\)\({}<s\)\({}\le1\)\({}+\frac d2\) and the minimal smoothness of \(G\) are not determined, and the constants are very large. This is an unrefereed note.