Tomas–Stein Extremizers on the Circle

Definitions and mathematical statement

Let $S^1=\{\omega\in{\mathbb{R}}^2:|\omega|=1\}$. For explicit notation, take $\sigma$ to be arclength measure and use the Fourier transform convention \[ \widehat{f\sigma}(x)=\int_{S^1}f(\omega)e^{-i x\cdot\omega}\,{\,\mathrm{d}}\sigma(\omega),\qquad x\in{\mathbb{R}}^2. \] These measure and Fourier conventions are stated here because the supplied record does not specify them; the question of whether constants attain the optimal norm is unaffected by the usual changes of these conventions.

For a scalar field $\mathbb K\in\{{\mathbb{R}},{\mathbb{C}}\}$, define the optimal Tomas--Stein extension constant by \[ C_{\mathbb K}=\sup_{0\ne f\in L^2(S^1,\sigma;\mathbb K)} \frac{\|\widehat{f\sigma}\|_{L^6({\mathbb{R}}^2)}}{\|f\|_{L^2(S^1,\sigma)}}. \] A nonzero function is an extremizer if its quotient equals $C_{\mathbb K}$.

Question. For the scalar field intended by the source, do nonzero constant functions attain this optimal constant? Equivalently, is \[ C_{\mathbb K}=\frac{\|\widehat{\sigma}\|_{L^6({\mathbb{R}}^2)}}{\sqrt{2\pi}}, \] or, equivalently, does every $f\in L^2(S^1,\sigma;\mathbb K)$ satisfy \[ \|\widehat{f\sigma}\|_{L^6({\mathbb{R}}^2)} \leq \frac{\|\widehat{\sigma}\|_{L^6({\mathbb{R}}^2)}}{\sqrt{2\pi}} \|f\|_{L^2(S^1,\sigma)}? \] The source does not explicitly specify real-valued versus complex-valued functions. The parameter $\mathbb K$ exposes that unresolved choice; this draft does not assert an equivalence between the two formulations. The question concerns attainment by constants, without asserting uniqueness or classifying all extremizers.

Qualifications from the supplied original excerpt. A function $g$ on $(S^1)^3$ is called antipodal when it is even in each argument separately: for each $j\in\{1,2,3\}$, replacing $\omega_j$ by $-\omega_j$ leaves $g$ unchanged almost everywhere. The excerpt discusses a reduction of the conjectured sharp inequality to positive semidefiniteness of a quadratic form on antipodal functions in $L^2((S^1)^3,\sigma^{\otimes3})$. Its corrupted transcription suggests the formal expression \[ \begin{aligned} Q(g)={}&\int_{(S^1)^6} \delta(\omega_1+\omega_2+\omega_3+\omega_4+\omega_5+\omega_6) \bigl(|\omega_1+\omega_2+\omega_3|^2-1\bigr)\\ &\quad\times\bigl(|g(\omega_1,\omega_2,\omega_3)|^2 -g(\omega_1,\omega_2,\omega_3)g(\omega_4,\omega_5,\omega_6)\bigr) \prod_{j=1}^{6}{\,\mathrm{d}}\sigma(\omega_j). \end{aligned} \] Here the $\delta$ notation is retained from the excerpt as a formal Dirac constraint. The supplied fragment does not provide a rigorous interpretation or domain for this singular expression, and it supplies no complex conjugate in the mixed product. Consequently, this display is a transcription aid, not a complete definition of a quadratic form on complex $L^2$, and no precise equivalence with the main question is asserted here.

The excerpt attributes to Barker, Thiele and Zorin-Kranich numerical verification of positive semidefiniteness on antipodal functions with Fourier modes up to degree $120$, and reports concentration of eigenfunctions associated with small eigenvalues in that finite-dimensional computation. The meaning of the degree cutoff is not specified. It also reports positive semidefiniteness on a certain unspecified subspace containing functions with localized support, with a corollary giving the conjectured sharp inequality for functions supported near a pair $\{p,-p\}$, $p\in S^1$. The printed neighborhood size contains $6/80$ and a detached square-root sign; neither its exact radius nor its distance convention can be recovered reliably from this record.

The supplied literature assessment, dated $2026\text{-}08\text{-}22$, attributes the localized sharp inequality to Becker and labels the overall problem partially solved, with the global constant-extremizer conjecture remaining open. These are supplied dated curation claims, not independently verified mathematical or literature conclusions.

Short English statement

Does a nonzero constant function maximize the ratio of the $L^6({\mathbb{R}}^2)$ norm of the circle extension operator to the $L^2(S^1,\sigma)$ norm of its input? The supplied record reports a localized partial result, while leaving the global question open; the scalar field and several details of the auxiliary quadratic-form formulation require review.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005543$, exact source problem code catalogue entry, title Tomas--Stein Extremizers on the Circle. The record gives proposed year $2023$ and names no proposer.
• Supplied report citation: Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory (2023), Oberwolfach Reports, pp. 1397--1452, DOI 10.4171/owr/2023/25. https://doi.org/10.4171/owr/2023/25. The supplied background describes these reports as documenting workshops and their open-problem discussions.
• Supplied literature source, labeled "OWR 2023 25": https://publications.mfo.de/bitstream/handle/mfo/4055/OWR_2023_25.pdf?sequence=1.
• Supplied literature source, labeled "Becker": https://arxiv.org/abs/2304.02345. The supplied assessment attributes a sharp inequality for functions localized near an antipodal pair to Becker; no full bibliographic description is supplied.
• Supplied literature source, labeled "Barker et al": https://arxiv.org/abs/2002.05118. The original excerpt attributes the numerical computations to Barker, Thiele and Zorin-Kranich and labels that reference [1]; it also refers to recent work [2], without supplying the numbered bibliography in the excerpt.
• Literature checked at: 2026-08-22. The supplied dated curation labels the problem partially_solved and states that the global constant-extremizer conjecture remains open. This assessment and its cited claims have not been independently verified here; the supplied background explicitly requests independent verification of time-sensitive status and claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is supplied verification metadata, not a verification performed for this draft.