Foschi Maximizers for Wave Strichartz Inequalities

Definitions and mathematical statement

Let $d\in{\mathbb{Z}}$, $d\geq2$, and put \[ \phi_d(x)=(1+|x|^2)^{-(d-1)/2},\qquad x\in{\mathbb{R}}^d. \] The wave evolution associated with the stated initial datum is \[ \begin{cases} \partial_t^2u-\Delta_xu=0 & \text{on }{\mathbb{R}}\times{\mathbb{R}}^d,\\ u(0,x)=\phi_d(x),\\ \partial_tu(0,x)=0. \end{cases} \] Equivalently, write $u_d(t)=\cos(t\sqrt{-\Delta})\phi_d$, where $\sqrt{-\Delta}$ is the spatial Fourier multiplier with symbol $|\xi|$.

Question attributed to the Foschi conjecture in the supplied record. For every integer $d\geq2$, does the datum $(\phi_d,0)$ attain equality in the sharp wave Strichartz inequality intended by the source?

Unresolved specification of the inequality. The supplied question does not state the spacetime exponent or norm, the initial-data space or norm, or the class of competing data. These choices are necessary to define a maximizer. More explicitly, if the intended inequality is \[ \|u\|_{Y_d}\leq C_d\|(u(0),\partial_tu(0))\|_{X_d}, \] where $X_d$ is the intended normed initial-data space, $Y_d$ is the intended spacetime normed space, and $C_d$ is the least valid constant, the requested assertion would be \[ \|u_d\|_{Y_d}=C_d\|(\phi_d,0)\|_{X_d} \qquad\text{for every integer }d\geq2. \] Here $X_d$ and $Y_d$ are placeholders for missing source specifications, not choices supplied by this draft. In particular, admissibility of the proposed datum in the intended space and the precise equality-case problem remain to be established from the source.

The original excerpt instead discusses an estimate for the positive cone \[ K_d^+=\{(\tau,\xi)\in{\mathbb{R}}\times{\mathbb{R}}^d:\tau=|\xi|\}, \] with measure $\mu_+$ defined by \[ \int_{K_d^+}F\,{\,\mathrm{d}}\mu_+ =\int_{{\mathbb{R}}^d}F(|\xi|,\xi)\,\frac{{\,\mathrm{d}}\xi}{2|\xi|}. \] Its displayed estimate has the form \[ \|\widehat{f\mu_+}\|_{L^q({\mathbb{R}}^{1+d})} \leq C_{d,p}\|f\|_{L^p(\mu_+)}. \] The excerpt says that this cone estimate generalizes an earlier inequality and that the Foschi conjecture is to be extended to that setting. However, the earlier inequality and the generalized maximizer conjecture are absent, and the displayed restrictions on $p,q$ are corrupted. Thus the excerpt does not determine the missing wave inequality or establish that its maximizer question is identical to the generalized cone question.

Short English statement

For every spatial dimension $d\geq2$, the supplied question asks whether the wave with initial displacement $(1+|x|^2)^{-(d-1)/2}$ and zero initial velocity attains the optimal Strichartz bound. The exact inequality and competing data class are missing, so a unique fully specified maximization problem cannot yet be stated. The supplied dated curation labels the problem partially solved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005158$, title Foschi Maximizers for Wave Strichartz Inequalities, exact source problem code catalogue entry. The record attributes the underlying conjecture to Foschi in its exposition but supplies no named proposer for this entry. Its proposed year is 2022.
• Supplied report citation: Real Analysis, Harmonic Analysis and Applications (2023), Oberwolfach Reports, pp. 1743--1803, DOI 10.4171/owr/2022/31, https://doi.org/10.4171/owr/2022/31. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The citation year 2023 and the record's proposed year 2022 are retained as supplied.
• Supplied literature source: Negro, exponentials rarely maximize cone extension, https://doi.org/10.1112/jlms.70112.
• Supplied literature source: Goncalves--Negro, local maximizers, https://arxiv.org/abs/2003.11955.
• Supplied literature source: OWR report, https://ems.press/content/serial-article-files/46969.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem partially solved and states that the generalized cone adjoint-restriction maximizer question has sharp global answers in low dimensions and local-maximizer results in broad dimensions, while the full global conjecture remains unresolved. This is supplied dated curation, not an independently verified assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and notation repaired. Those are supplied verification claims; this draft does not independently verify them or the cited sources.