Critical boundedness problem for the Born–Jordan distribution

The problem concerns the endpoint mapping question for the Born–Jordan distribution, raised by Stra, Svela, and Trapasso. A May 11, 2026 preprint had identified the higher-dimensional critical case as the only unresolved one; a later preprint now claims a complete settlement, including sharp Lorentz-space bounds.

Known results

• For dimension $d \ge 2$, the critical exponent was identified as $p_{*}(d)=\frac{2d}{d-2}$, with $p_{*}(2)=\infty$.
• The subcritical regime $1\le p<p_{*}(d)$ was proved finite and attained.
• For $d=2$, the critical case $p=\infty$ was proved unbounded; for $d>2$, the supercritical regime $p>p_{*}(d)$ was also proved unbounded.
• Before the newer claim, the critical case $p=p_{*}(d)$ for $d>2$ remained open and was equivalent by duality to the endpoint $q=\frac{2d}{d+2}$ mapping question.

September 8, 2026 claimed settlement

The preprint Endpoint bounds for the Born-Jordan distribution claims to settle the critical boundedness problem and establish Lorentz sharpness, resolving the endpoint mapping question. It is an unrefereed preprint, so the claimed proof and its exact scope remain unverified.

Current status (as of September 2026): the previously open higher-dimensional critical endpoint is claimed solved by an unrefereed preprint, while independent mathematical verification remains outstanding.