Kernels and Involutivity of Exotic Fourier Transforms

Definitions and mathematical statement

The supplied record denotes an exotic Fourier transform by $F^G_\rho$, its defining kernel by $\phi^G_\rho$, and its underlying function domain by $G^F$. It does not define $G$, $F$, $\rho$, or $G^F$, specify the functions under consideration or their scalar field, or give the rule relating the kernel to the transform. These symbols therefore retain their source meanings without additional assumptions in this draft.

Requested computation and extension. Compute explicitly the kernel $\phi^G_\rho$ defining $F^G_\rho$ on the intended space of functions on $G^F$, and extend $F^G_\rho$ to an involutive operator.

For clarity, an involutive operator on a space $W$ is a map $T\colon W\to W$ satisfying \[ T\circ T=\operatorname{id}_W, \qquad\text{equivalently}\qquad T(T(w))=w\quad\text{for every }w\in W. \] If the intended extension is by inclusion of an original domain $V\subseteq W$, agreement would mean $T(v)=F^G_\rho(v)$ for every $v\in V$, with the original outputs identified in $W$. This describes only that possible meaning of extension: the source does not specify $V$, $W$, the required identifications, or whether this is the intended construction.

Unresolved qualifications. No kernel formula, normalization, admissible class of data, or target space for the extension is supplied. Consequently, the record supports the requested task above, but does not determine a fully specified computational problem or a quantified existence assertion.

Short English statement

The problem asks for an explicit kernel for an exotic Fourier transform and an extension whose square is the identity. The supplied excerpt leaves the transform, its parameters, its function space, and the extension space undefined, so a precise formulation requires further source context. The supplied open label is a dated curation claim, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004787$, source problem code catalogue entry, title "Kernels and Involutivity of Exotic Fourier Transforms." The supplied record contains no named proposer and lists the proposed year as 2021.
• Supplied report citation: Automorphic Forms, Geometry and Arithmetic (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/39, pp. 2089--2156. Supplied source URL and literature reference labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2021/39. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The report contents have not been independently consulted for this draft.
• Supplied literature assessment, checked exactly 2026-08-22: the record labels the problem open and describes the extracted passage as an incomplete computational problem because the operator and requested formula are unspecified; it consequently reports that no identifiable resolution could be verified. This is supplied dated curation and literature triage, not an independently verified assessment. The supplied background calls for independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note states that explicit formulas were dollar-delimited and the supplied mathematical content was preserved. This records a formatting and preservation check, not verification of mathematical completeness or correctness.