Fractal Local Smoothing for General Time Sets

Definitions and mathematical statement

Let $E\subset[1,2]$ be an arbitrary time set. The supplied excerpt defines $N(E,\delta)$, for $\delta>0$, as the minimum number of intervals of length $\delta$ required to cover $E$. It also mentions intervals $J\subset[1,2]$ satisfying $|J|\geq\delta$, but the formula in which their supremum occurs is missing.

The intended conjecture asks whether, for every such $E$, fractal local smoothing holds throughout the parameter range \[ p>2,\qquad s>\frac{1}{p}\nu_E^{\#}(p s_p). \] More specifically, the supplied original excerpt asks whether its inequality $\text{(3)}$ holds with the sum taken over $t\in E_j$ whenever these strict inequalities hold.

Unresolved formulation. Inequality $\text{(3)}$ is absent from the supplied record. The record does not define the frequency or scale parameter $j$, the sets $E_j$, the exponent $s_p$, or the Legendre--Assouad function $\nu_E^{\#}$. It also does not specify the operator, spatial dimension, function class, norms, scale factors, or permitted dependence of constants in the proposed estimate. Consequently, the displayed parameter range records the intended target, but does not by itself specify a complete mathematical inequality. No definitions or missing hypotheses are inferred here.

The excerpt describes the Assouad spectrum as $\theta\mapsto\dim_{A,\theta}E$, defined through an infimum over exponents $a>0$ for which a constant $C$ exists, but the defining covering inequality and the domain of $\theta$ are absent. It also mentions $\alpha\in{\mathbb{R}}$ without its defining formula. Thus these passages do not supply a definition of $\nu_E^{\#}$.

The source attributes to its reference [2] the corresponding estimate for radial functions and failure when \[ s<\frac{1}{p}\nu_E^{\#}(p s_p). \] These are supplied background assertions whose precise hypotheses and meanings cannot be recovered from the excerpt. The source further states that $E=[1,2]$ recovers its classical local smoothing conjecture $\text{(2)}$, which is also absent. Neither assertion resolves the equality case $s=p^{-1}\nu_E^{\#}(p s_p)$, and the requested conjecture uses a strict upper-side inequality.

Short English statement

The problem asks whether the fractal local smoothing estimate holds for every time set $E\subset[1,2]$, every exponent $p>2$, and every regularity parameter $s>p^{-1}\nu_E^{\#}(p s_p)$. The supplied record omits the estimate and essential definitions, so a complete precise formulation requires human review of the cited source. The open status and the radial and failure assertions are supplied source claims, not independently verified conclusions.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30006374$, title Fractal Local Smoothing for General Time Sets, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2025.
• 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 background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The cited report text was not independently consulted.
• Supplied literature source, labelled "Beltran--Roos--Rutar--Seeger primary paper": https://arxiv.org/abs/2501.12805.
• 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 and states that the general-time-set conjecture remains unresolved, with radial and structural partial results rather than the stated general bound. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The excerpt cites [2] for the radial estimate and the failure assertion, and [3], [12], and [1] for the origin of the relevant function in sharp $L^p$-improving bounds for spherical maximal functions with fractal dilation sets. Their bibliography entries and their correspondence with the supplied links are not given.