Kwaśnicki’s uniform eigenfunction bound conjecture and the Kaleta–Kwaśnicki–Małecki conjectural remainder

Let $\alpha\in(0,2)$ , and let $\{\varphi_n^{(\alpha)}\}_{n\geq 1}$ be an $L^2((-1,1))$ -normalized sequence of eigenfunctions of the fractional Laplacian on $(-1,1)$ with the appropriate exterior condition. Is there a constant $C<\infty$ , independent of $\alpha$ and $n$ , such that $\|\varphi_n^{(\alpha)}\|_{L^\infty((-1,1))}\leq C$ for every $\alpha\in(0,2)$ and every $n\geq 1$ ?

References

Primary source

arXiv

Additional references

Eigenvalue asymptotics and uniform eigenfunction bounds for the fractional Laplacian in the interval — arXiv

fractional-laplacians partial-differential-equations spectral-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to settle both fractional-Laplacian questions, but the result has not yet been independently verified.

The problem concerns a uniform bound for eigenfunctions and a precise remainder term in fractional-Laplacian eigenvalue asymptotics. No proposer or original date is identified in the retrieved material.

August 24, 2026 preprint

The preprint claims the conjectural eigenvalue remainder and proves the numerically suggested uniform eigenfunction bound, which would settle both questions. Its claims are unrefereed and remain unverified.

Current status (as of August 2026): Both conjectures are claimed solved by an unrefereed preprint, but independent verification is still pending.

Sources

No solutions have been posted yet.