Legendre–Assouad Bounds for Fractal Local Smoothing

Definitions and mathematical statement

Let $d$ denote the spatial dimension and let $\Delta=\sum_{k=1}^{d}\partial_{x_k}^{2}$ be the Laplacian on ${\mathbb{R}}^d$. The half-wave operator $e^{it\sqrt{-\Delta}}$ has Fourier multiplier $e^{it|\xi|}$. For a real exponent $p>2$, put \[ s_p=(d-1)\left(\frac12-\frac1p\right). \] The record refers to a time set $E$, discrete time sets $E_j$, frequency projections $P_j$, and a function $\nu_E^{\#}$ called the Legendre--Assouad function. Their precise definitions and admissibility conditions are not supplied.

Under the intended uniform-estimate interpretation, the proposed assertion is: for every real $p>2$ and every real $s$ satisfying \[ s>\frac1p\nu_E^{\#}(p s_p), \] there exists a finite constant $C$ such that \[ \left(\sum_{t\in E_j} \|e^{it\sqrt{-\Delta}}P_jf\|_{L^p({\mathbb{R}}^d)}^p\right)^{1/p} \le C\,2^{js}\|f\|_{L^p({\mathbb{R}}^d)} \] for every admissible frequency index $j$ and every admissible function $f$, with $C$ independent of $j$ and $f$. The admissible dimensions, time sets, index range, function class, and other permitted dependencies of $C$ require confirmation from the source. In particular, neither the construction nor the spacing or covering properties of $E_j$ may be inferred from the supplied record.

The original transcription contains an incomplete candidate definition of the form \[ \nu_E^{\#}(\alpha) =\lim_{\delta\to0} \frac{\log\!\left(\sup_{\delta\le |J|\le1} |J|^{-a}N(E\cap J,\delta)\right)} {\log(1/\delta)}. \] This is reproduced only as an unresolved transcription fragment, not as an established definition. The exponent $a$ is not identified with the argument $\alpha$; the covering number $N$, the class of $J$, the allowed time sets, and the existence of the displayed limit are unspecified. Whether a limit superior is intended also requires verification. Consequently, the supplied evidence does not determine a fully specified conjecture.

The supplied source account states that the estimate holds for radial functions and fails below the proposed threshold $s<p^{-1}\nu_E^{\#}(p s_p)$, and that setting $E=[1,2]$ recovers the classical local smoothing conjecture. These are supplied claims with unresolved underlying hypotheses, not independently verified results. No assertion at the boundary $s=p^{-1}\nu_E^{\#}(p s_p)$ is supplied.

Short English statement

The conjecture asks whether a frequency-localized half-wave evolution, sampled at times in $E_j$, satisfies the displayed $L^p$ estimate whenever $s$ exceeds the threshold determined by the Legendre--Assouad function of $E$. Essential definitions and hypotheses are missing or corrupted in the supplied transcription. The supplied dated assessment labels the general estimate open and reports radial validity and sharpness below the threshold; these claims require source verification and human mathematical review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30006373$, title "Legendre--Assouad Bounds for Fractal Local Smoothing," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year 2025 and no named proposer.
• Supplied report citation: Real Analysis, Harmonic Analysis, and Applications (2026), Oberwolfach Reports, pp. 1707--1752, DOI 10.4171/owr/2025/32. https://doi.org/10.4171/owr/2025/32. The citation year 2026 differs from the proposed year and the year appearing in the DOI; this discrepancy is unresolved.
• Supplied literature source, labelled "Beltran--Roos--Rutar--Seeger primary paper": https://arxiv.org/abs/2501.12805. The original excerpt attributes the radial result and failure below the threshold to its reference [2], but does not supply the bibliography needed to identify [2] conclusively.
• Supplied literature source, labelled "published fractal local smoothing paper": https://doi.org/10.1112/blms.70214.
• Supplied literature source, labelled "Oberwolfach report": https://doi.org/10.4171/owr/2025/32.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports the conjecture for arbitrary time sets as proved for radial functions and sharp below the predicted exponent, and says the general upper estimate remains open. This is supplied dated curation, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.