Weber–Zacher conjecture on the fractional Li–Yau constant

The conjecture asks whether the optimal fractional Li–Yau constant tends to the classical value as the fractional order approaches $2$. Earlier work established the relevant inequality but explicitly left this limiting question open.

Known results

• Weber and Zacher established a positive constant depending only on $\beta$ and $d$ for the fractional heat equation.
• Li, 2020, proved the fractional Li–Yau inequality and gave an explicit expression when $\beta=1$, while noting that the limit as $\beta\to2$ remained open.

September 2026 claimed resolution

Huaiqian Li reports bounds $d/\beta\le C_{LY}(\beta,d)\le d/\beta+K_d(2-\beta)\log\!\left(e/(2-\beta)\right)$, which imply $C_{LY}(\beta,d)\to d/2$ as $\beta\uparrow2$. This addresses the conjectured fixed-dimension limit, but the result remains unverified in this scan.

Current status (as of September 2026): The fractional Li–Yau inequality is established, and a recent preprint claims the conjectured limit $C_{LY}(\beta,d)\to d/2$ as $\beta\uparrow2$; independent verification is not recorded.