Weighted Sobolev Stability Under Nonlinear Composition

Definitions and mathematical statement

For a coefficient family $a=\{a_{\widetilde{\ell}}(t)\}_{\widetilde{\ell}\in{\mathbb{N}}^2}$ on the source interval $I=[t_-,t_+]$, and a real number $s>1$, define the possibly infinite quantity \[ \mathcal{B}_s(a):=\sum_{\widetilde{\ell}\in{\mathbb{N}}^2}(1+|\widetilde{\ell}|)^{2s}\sup_{t\in I}|a_{\widetilde{\ell}}(t)|^2. \] The conventions for ${\mathbb{N}}$, the index size $|\widetilde{\ell}|$, and the absolute value on coefficient values are not specified in the supplied record.

Let $G$ be smooth with a first-order zero at the origin. In the usual smooth-map interpretation, this means $G(0)=0$ and $DG(0)\ne0$; the domain, codomain, and intended derivative convention are not supplied.

The source asks whether, for every such $G$ and every real $s>1$, there exists a finite constant $C_2=C_2(G,s)\ge0$ such that every coefficient family $\phi=\{\phi_{\widetilde{\ell}}(t)\}_{\widetilde{\ell}\in{\mathbb{N}}^2}$ on $I$ satisfies \[ \mathcal{B}_s(\phi)\le1 \quad\Longrightarrow\quad \sum_{\widetilde{\ell}\in{\mathbb{N}}^2}(1+|\widetilde{\ell}|)^{2s}\sup_{t\in I}|(G\circ\phi)_{\widetilde{\ell}}(t)|^2\le C_2(G,s). \] Here $(G\circ\phi)_{\widetilde{\ell}}(t)$ is retained as source notation. The supplied record does not define how a coefficient family represents a function, which spatial domain and basis are used, or how the coefficients of its nonlinear composition are obtained. In particular, it does not establish that composition is defined for every family in the stated unit ball. Thus this is the extracted question with unresolved qualifications, rather than a fully specified assertion. No dependence of $C_2$ on the coefficient family or time interval is displayed in the source; whether the interval is fixed or the bound is intended to hold uniformly over intervals is unspecified.

Short English statement

The question asks whether smooth nonlinear composition by a function vanishing to first order at the origin preserves a bounded weighted coefficient norm with a supremum over time, for every $s>1$. The coefficient representation and the domain of composition must be specified before the proposed estimate becomes a precise mathematical assertion.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30004800$, entitled Weighted Sobolev Stability Under Nonlinear Composition, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Supplied report citation: Mathematical Aspects of General Relativity (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/40, pp. 2157--2267. Supplied source URL: https://doi.org/10.4171/owr/2021/40. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2021/40.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that the extracted question concerns a nonlinear Sobolev estimate near a first-order zero, that the report supplies the question without a proof, and that no authoritative general resolution was found. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that the explicit question was isolated from surrounding report prose and its notation normalized. This supplied editorial label does not establish mathematical correctness or resolve the missing definitions. The proposed year and year tag are $2021$, while the supplied report title includes $2022$; these details are retained without reconciliation.