Maz’ya’s Open Problem 9 on higher-order uncertainty principles

For each dimension $N\ge 2$ and higher order $m\ge 2$ , determine the sharp constants and all extremal divergence-free vector fields in the corresponding higher-order Heisenberg uncertainty inequalities. In particular, for vector fields $u$ on $\mathbb{R}^N$ satisfying $\operatorname{div}u=0$ , determine the optimal constant in Maz’ya’s higher-order solenoidal uncertainty inequality and characterize the cases of equality. The supplied sources do not state the precise higher-order differential functional and normalization used in the problem. The case $N=2$ is claimed in the cited preprint, while the general problem remains unresolved.

References

Primary source

arXiv

Additional references

Sharp higher-order uncertainty principles — arXiv — Hongtao Hu, Meiqi Liu, Wenming Zou

functional-analysis harmonic-analysis partial-differential-equations

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

A new unrefereed preprint claims to settle the two-dimensional case, while the full higher-order problem remains open.

Maz’ya’s problem asks for sharp higher-order uncertainty constants and extremals in a divergence-free setting. The latest claim concerns only the case $N=2$ ; no source here settles every formulation or dimension.

Known results

The sharp second-order scalar constant is $(N+2)^2/4$ , with Gaussian extremals.
For $N=2$ , rotated gradients reduce the divergence-free problem to the scalar second-order inequality, yielding optimal constant $4$ .
A later paper reports the sharp $L^2$ solenoidal constant in every dimension as $\frac14\left(\sqrt{(N-2)^2+8}+2\right)^2$ , with extremals described by dimension.

October 2026 two-dimensional claim

In an October 2026 preprint, Hongtao Hu, Meiqi Liu, and Wenming Zou claim sharp higher-order inequalities, optimal constants, and extremal functions that explicitly answer Maz’ya’s problem for $N=2$ . This is an unrefereed claim and does not close the broader problem.

Current status (as of October 2026): The $N=2$ case is claimed solved by an unrefereed preprint, while the general higher-order problem remains open.

Sources

No solutions have been posted yet.