Fractal-Time Local Smoothing

Definitions and mathematical statement

Let $E\subset[1,2]$ be the intended time set. For a scale index $j$, a maximal $2^{-j}$-separated subset $E_j\subset E$ means a subset satisfying $|t-u|\geq 2^{-j}$ for distinct $t,u\in E_j$, maximal with respect to inclusion. Such a subset is finite. The source calls $E$ fractal but supplies no dimension, covering estimate, or regularity hypothesis.

On ${\mathbb{R}}^d$, the half-wave operator is the Fourier multiplier defined by \[ \widehat{e^{it\sqrt{-\Delta}}g}(\xi)=e^{it|\xi|}\widehat g(\xi). \] The notation $P_j$ is interpreted conventionally as a smooth Fourier localization to frequencies $|\xi|\asymp 2^j$. For example, after fixing a smooth function $\varphi$ compactly supported away from the origin, one may define \[ \widehat{P_jf}(\xi)=\varphi(2^{-j}\xi)\widehat f(\xi). \] This cutoff convention is explanatory; the supplied record does not specify the multiplier defining $P_j$.

Extracted estimate and unresolved target. Fix a spatial dimension $d$, an exponent $p$, a real smoothness exponent $s$, an intended set $J\subset{\mathbb{Z}}$ of scale indices, and a choice of maximal separated sets $(E_j)_{j\in J}$. The estimate to be investigated is the existence of a finite constant $C$, independent of $j$ and $f$, such that \[ \left(\sum_{t\in E_j} \left\|e^{it\sqrt{-\Delta}}P_jf\right\|_{L^p({\mathbb{R}}^d)}^p\right)^{1/p} \leq C\,2^{js}\|f\|_{L^p({\mathbb{R}}^d)} \qquad(j\in J). \] A conventional test-function formulation takes every $f\in\mathcal S({\mathbb{R}}^d)$, the space of smooth functions whose derivatives decay faster than every inverse power of $|x|$. The displayed sum is unnormalized, as in the supplied clean statement. The damaged original excerpt mentions $2\leq p<\infty$, but does not reliably specify the full parameter scope of the fractal-time conjecture.

The requested task is to determine the sharp admissible smoothness exponents and parameter ranges for the intended class of time sets. The supplied record does not state that class, the relevant fractal parameters, a proposed sharp threshold, or endpoint conventions. It also does not specify the intended dimensions, scale range, or uniformity of $C$ over time sets and choices of maximal separated subsets. Thus the display specifies an estimate to investigate, but the supplied evidence does not determine a complete sharp conjecture.

For background only, the original excerpt introduces \[ s_p=(d-1)\left(\frac12-\frac1p\right), \qquad \sigma_p=\max\left\{0,s_p-\frac1p\right\}, \] and describes the classical continuous-time local smoothing conjecture using $s>\sigma_p$ for $p>2$. That background threshold is not supplied as the threshold for the unnormalized fractal-time sum above. The original excerpt has disrupted formula order and missing summation information, so its isolated fixed-time wording cannot resolve the missing fractal-time hypotheses.

Short English statement

Determine the sharp frequency growth in an $L^p({\mathbb{R}}^d)$ estimate for the half-wave propagator sampled at separated times in a fractal subset of $[1,2]$. The supplied record omits the fractal hypotheses and sharp exponent range, so this is an incomplete conjecture template requiring mathematical review. Its supplied status is partially_solved; that status is not independently verified.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006372$, title Fractal-Time Local Smoothing, 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, pp. 1707--1752, DOI https://doi.org/10.4171/owr/2025/32. The original excerpt includes the page marker 1745. The supplied citation year is 2026, while the DOI and source-year metadata refer to 2025; this discrepancy is unresolved.
• 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.
• Supplied literature source, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2025/32.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that classical local smoothing is settled in dimension two and substantial higher-dimensional ranges, whereas the extracted fractal-time extension is known only in restricted cases, including radial data. This is supplied dated curation, not an independently verified literature assessment; the record expressly requests independent verification of time-sensitive status and cited claims.
• The damaged original excerpt attributes classical local smoothing to Sogge [14], large-$p$ progress to Wolff [15], the range $p\geq 2(d+1)/(d-1)$ to Bourgain and Demeter [4], and the two-dimensional result for all $p>2$ to Guth, Wang and Zhang [8]. These are source-reported background attributions. Full bibliographic entries for these numbered references are not supplied.