Bilinear Wave–Schrödinger Restriction Without Transversality

Definitions and mathematical statement

For a spatial dimension $d$, write $\Delta_x=\sum_{j=1}^d\partial_{x_j}^2$. A free Schrödinger wave satisfies $(i\partial_t+\Delta_x)u=0$, and a free wave satisfies $(\partial_t^2-\Delta_x)v=0$. These definitions describe the interaction named in the supplied statement; the record does not determine whether the intended wave factor is a full wave or a particular half-wave component.

The supplied problem asks: determine the admissible exponents and the sharp dependence on localization scales in bilinear restriction estimates for wave--Schrödinger interactions after removing the transversality assumption. Here an admissible choice must make the intended bilinear estimate hold uniformly over its prescribed class of inputs, and sharp scale dependence means optimal dependence of its constant on the prescribed localization parameters. The record does not supply that estimate, its input and output norms, the localization parameters, or the transversality condition being removed. Consequently, this is an explicitly incomplete formulation of the requested target, and no particular quantified inequality or exponent range can be asserted from the supplied evidence.

The quoted background concerns a related transference result for free Schrödinger waves in four spatial dimensions, on ${\mathbb{R}}^{1+4}$, extending estimates to an endpoint inhomogeneous Strichartz space $Z$. The displayed norm in the supplied excerpt is corrupted. It contains the identifiable term $\|u\|_{L_t^\infty L_x^2({\mathbb{R}}^{1+4})}$ and a forcing term involving $(i\partial_t+\Delta_x)u$, but the complete forcing norm cannot be recovered unambiguously. Thus $Z$ is not fully defined here, and neither the four-dimensional setting nor this particular solution space is silently imposed on the missing wave--Schrödinger estimate. The excerpt describes a result, rather than stating the requested nontransverse conjecture. The supplied curation connects the question with regularity theory for the Zakharov equation without specifying a corresponding regularity threshold.

Short English statement

The question asks which bilinear wave--Schrödinger estimates survive without transversality, with optimal exponents and scale dependence. The supplied excerpt omits the actual inequality and contains a damaged definition of the endpoint space, so a complete precise conjecture requires further source recovery and human mathematical review. Its open status is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30004843$, exact source problem code catalogue entry; supplied complete source record titled "Bilinear Wave--Schrödinger Restriction Without Transversality."
• Supplied report citation: MATRIX-MFO Tandem Workshop/Small Collaboration: Rough Wave Equations (2022), Oberwolfach Reports, pp. 2459--2496, DOI 10.4171/owr/2021/45. Source and literature-source link, labeled "Oberwolfach Report 45": https://doi.org/10.4171/owr/2021/45.
• Supplied literature source: "Transference of bilinear restriction estimates," https://arxiv.org/abs/1605.04882.
• Supplied literature source: "Energy-critical Zakharov system," https://arxiv.org/abs/2005.09932.
• Attribution: the quoted excerpt describes the first result as joint work with S. Herr and K. Nakanishi, referring to its local reference [3]. The speaker and the bibliographic identity of [3] are not supplied; no proposer is identified.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports that a transference principle establishes bilinear restriction estimates for the stated endpoint space in an available transverse setting, and describes the sharp nontransverse estimate as open and connected with Zakharov regularity. This is supplied dated curation, not independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that an earlier inspection of the linked report recovered an open question following a sharp transverse estimate. This is a supplied source-verification claim; no source inspection was performed for this draft. The registry gives proposed year 2021, while its report citation gives 2022; both are retained without reconciliation.