Fourier Uniqueness from Vanishing on General Surfaces

Definitions and mathematical statement

For a specified Fourier setting, let $X$ be a class of test functions, let $\widehat f$ denote the Fourier transform of $f\in X$, and let $A$ and $B$ be prescribed subsets of the respective domains of $f$ and $\widehat f$. Assume that the restrictions below are defined. Call $(A,B)$ a Fourier uniqueness pair for $X$ if \[ \forall f\in X,\qquad \left(f|_A=0\ \text{and}\ \widehat f|_B=0\right) \Longrightarrow f=0. \] Here equality to zero and restriction must be interpreted in the chosen function class; the source does not supply that interpretation.

Source question. In Fourier uniqueness problems where the prescribed vanishing sets are spheres, do analogous uniqueness results, or analogous integration-by-parts results, hold when those vanishing sets are replaced by other surfaces?

For the uniqueness component, the displayed implication gives a conditional framework: identify admissible replacements $A,B$ by other surfaces, or collections of surfaces, and determine hypotheses under which the implication holds for the intended class $X$. The supplied record does not specify the ambient space or dimension, the test-function class, the Fourier-transform convention, the original spheres or their radii, the permitted replacement surfaces, or the quantifiers governing those replacements. Consequently, this framework is not a fully specified conjecture about all surfaces. The integration-by-parts component is also unresolved at the level of formulation: no baseline identity, boundary terms, regularity assumptions, or proposed analogue is supplied.

The supplied dated curation labels the problem partially solved and reports that uniqueness has been proved for concentric spherical shells and several perturbations of those spheres, while the broad question concerning arbitrary other surfaces remains open. These are supplied literature assessments, not independently verified mathematical or current-status claims; their precise hypotheses are absent from this record.

Short English statement

The question asks when vanishing of both a test function and its Fourier transform on prescribed nonspherical surfaces forces the function to be zero, and whether related integration-by-parts results extend to such surfaces. The source leaves the function class, geometric hypotheses, quantifiers, and intended integration-by-parts statement unspecified, so a precise conjecture requires further source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004508$, title Fourier Uniqueness from Vanishing on General Surfaces, in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. The record lists proposed year $2020$ and supplies no named proposer.
• Supplied report attribution: Automorphic Forms and Arithmetic (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/26, pp. 1317--1383. https://doi.org/10.4171/owr/2020/26. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No exact page or passage for this question is supplied.
• Supplied literature source: Fourier interpolation from spheres. https://arxiv.org/abs/2002.11627.
• Supplied literature source: Perturbed Fourier uniqueness and interpolation. https://arxiv.org/abs/2103.12015.
• Supplied literature source: Fourier uniqueness in even dimensions. https://doi.org/10.1073/pnas.2023227118.
• Literature assessment date: 2026-08-22. This is supplied dated curation derived from the supplied OWR curation, not an independently verified assessment. It labels the problem partially solved and reports uniqueness for concentric spherical shells and several perturbations, with the broad question about arbitrary other surfaces remaining open. The supplied record expressly calls for independent verification of time-sensitive status and cited claims.