Fractal-Time Local Smoothing Between Lebesgue Spaces

Definitions and mathematical statement

Let $p,q$ be real exponents satisfying $1<p\leq q<\infty$, and define the Hölder conjugate $p'=p/(p-1)$. Assume $q>p'$. Write $\Delta=\sum_{k=1}^{d}\partial_{x_k}^{2}$ for the Laplacian on ${\mathbb{R}}^d$. The half-wave operator is conventionally defined, initially on Schwartz functions, by \[ \widehat{e^{it\sqrt{-\Delta}}g}(\xi)=e^{it|\xi|}\widehat g(\xi). \] The supplied record does not specify the admissible dimensions $d$, the time set $E$, the construction of the sets $E_j$, the operators $P_j$, or the function $\nu_E^{\#}$. These symbols are therefore retained as unresolved source notation; no definitions or admissibility hypotheses for them are imposed here.

With those qualifications, the intended conjecture asks whether, for every pair $p,q$ as above and every real $s$ satisfying \[ s>\frac{d+1}{2}\left(\frac1p-\frac1q\right) +\frac1q\nu_E^{\#}\!\left(\frac{q(d-1)}2\left(1-\frac1p-\frac1q\right)\right), \] there is a finite constant $C$, independent of the integer $j\geq1$ and the input $f$, such that \[ \left(\sum_{t\in E_j} \left\lVert e^{it\sqrt{-\Delta}}P_jf\right\rVert_{L^q({\mathbb{R}}^d)}^q\right)^{1/q} \leq C\,2^{js}\lVert f\rVert_{L^p({\mathbb{R}}^d)}. \] The intended input space is $L^p({\mathbb{R}}^d)$; the meaning of the composed operator on that space depends on the missing definition of $P_j$. The time sum is the unweighted sum displayed in the source. No assertion at equality in the condition on $s$, or at $q=p'$, is supplied. The permitted dependence of $C$ on the time set and its discretizations is unspecified. Thus this is a conditional transcription of the requested estimate, rather than a fully specified conjecture.

The supplied exposition states that the threshold matches lower bounds proved in its reference $[2]$, that the conjecture is verified in the easy case $p=2$, and that a solution would immediately imply a conjecture about closures of $L^p\to L^q$ type sets for spherical maximal functions with fractal dilation sets. These are supplied claims, not independently verified conclusions; neither reference $[2]$ nor the latter conjecture is fully identified in the excerpt.

Short English statement

The question asks for a uniform frequency-dependent $L^p({\mathbb{R}}^d)$-to-$\ell^q(L^q({\mathbb{R}}^d))$ estimate for half-wave evolution sampled at fractal times, with the displayed strict lower bound on $s$. The supplied record labels the general conjecture open and reports a known $p=2$ case. Missing definitions of the time sets, frequency operators, and fractal invariant prevent a fully precise formulation.

Sources

• Local UnsolvedMath registry: the supplied complete source record for permanent ID $30006375$, titled Fractal-Time Local Smoothing Between Lebesgue Spaces, exact 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, DOI 10.4171/owr/2025/32, pp. 1707--1752. https://doi.org/10.4171/owr/2025/32. This is also a supplied literature source. The citation's year 2026 and the year 2025 in the DOI and registry metadata are retained without reconciliation.
• Supplied literature source, labeled "Beltran--Roos--Rutar--Seeger primary paper": https://arxiv.org/abs/2501.12805.
• Supplied literature source, labeled "published fractal local smoothing paper": https://doi.org/10.1112/blms.70214.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports matching lower bounds and verification for the easy case $p=2$, and describes the general range as conjectural. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and states that the official report was checked and its notation repaired. Those are supplied verification claims; no source visit or independent verification is asserted in this draft. The excerpt's reference $[2]$ has no accompanying bibliography entry establishing its identity.