Near-sharp trilinear convolution range on transverse patches of the 3-sphere

Definitions and mathematical statement

Bez's AIM question asks for all exponents $p,q>0$ such that, for three transverse patches $S_j\subset S^3\subset{\mathbb{R}}^4$ with surface measures $\sigma_j$, \[ \|g_1\,d\sigma_1*g_2\,d\sigma_2*g_3\,d\sigma_3\|_{L^q({\mathbb{R}}^4)} \leq C\prod_{j=1}^3\|g_j\|_{L^p(S_j,d\sigma_j)}. \] Convolution is in the ambient ${\mathbb{R}}^4$; for $q<1$ the displayed expression is its usual quasi-norm. The supplied report uses compact positive-area patches with quantitative transversality $\|\nu_1\wedge\nu_2\wedge\nu_3\|\geq c_0>0$ for all triples of normals. Constants may depend on fixed patch geometry and exponents. The source's damaged comparison sign is reconstructed as this boundedness question, not as an assertion that every exponent works.

The report claims the exact low-target-exponent range \[ 0<q\leq1:\qquad\text{the estimate holds exactly when }p\geq1. \] For $1<q\leq\infty$, with $1/\infty=0$, it claims sufficiency throughout \[ \frac{15}{p}<8+\frac7q \] and failure throughout the exterior region \[ \frac{15}{p}>8+\frac7q. \] Consequently its only unresolved strong-type exponents are \[ 1<q\leq\infty,\qquad\frac{15}{p}=8+\frac7q. \] This includes an unresolved $L^\infty$ boundary at $p=15/8$ in the report. Boundary weak-type or restricted-type conclusions are not supplied and are not inferred.

The stated sufficiency input is Bejenaru's 2022 sharp-up-to-endpoint trilinear restriction theorem with parameters $k=3,n=4$ at every extension-product exponent $s>14/15$. The report combines it with a characteristic-function Fourier $L^1$ estimate, factorized Lorentz summation and interpolation. Its necessary conditions come from a one-cap test and a squashed-cap construction. None of the complete theorem hypotheses, conversion lemmas or test-family details are present in the supplied record. These remain attributed partial-progress claims, not an independently verified resolution of the strong boundary.

Short English statement

Determine the trilinear singular-convolution range on transverse sphere patches. The report settles target quasi-norm exponents up to one, and gives matching strict sufficiency and exterior failure for larger target exponents. Its claimed remaining question is the strong-type line $15/p=8+7/q$ for $q>1$, including the infinite-target endpoint.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent ID $20000537$, source code catalogue entry; damaged original equation (0.12), supplied verification and research summary.
• AIM workshop "Carleson theorems and multilinear operators", Question 10, attributed to Bez: https://aimath.org/pastworkshops/multilinopsproblems.pdf. Canonical location aim-analysis-notes.json, notes[68]; original remarks and literature field empty. The supplied verification reports a TeX/PDF check; no such source was opened here.
• Supplied attribution: Bejenaru's 2022 trilinear restriction theorem for $k=3,n=4$, at exponents $s>14/15$. No title, identifier, full theorem or conversion-lemma references are provided.
• Supplied machine-generated research attempt 1: partial result; problem partially solved at the run and in the registry; rigor metadata "proved", novelty confidence low. The non-peer-reviewed strict-region synthesis requires independent verification before citation.