Let Omega_n=-1,1^n with normalized uniform measure, and let nabla and Delta denote the Walsh gradient and Walsh Laplacian, respectively. The conjecture asks whether, for every 1<p≤ 2, there is a constant C<∞ independent of n such that for every n∈N and every function f on Omega_n, |nabla f|_L_p(Omega_n;ell_2^n)≤ C(p-1)^-2|Delta^1/pf|_L_p(Omega_n). The fractional exponent 1/p is conjectured to be optimal.

Sharp fractional Riesz estimate on the hypercube

Let $\Omega_n=\{-1,1\}^n$ with normalized uniform measure, and let $\nabla$ and $\Delta$ denote the Walsh gradient and Walsh Laplacian, respectively. The conjecture asks whether, for every $1<p\le 2$ , there is a constant $C<\infty$ independent of $n$ such that for every $n\in\mathbb{N}$ and every function $f$ on $\Omega_n$ , $\|\nabla f\|_{L_p(\Omega_n;\ell_2^n)}\le C(p-1)^{-2}\|\Delta^{1/p}f\|_{L_p(\Omega_n)}$ . The fractional exponent $1/p$ is conjectured to be optimal.

References

Primary source

arXiv

Additional references

Sharp Fractional Riesz Estimates on the Hypercube — arXiv

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 3, 2026 unrefereed preprint claims to settle the sharp fractional Riesz estimate on the hypercube.

The problem concerns the conjectured optimal exponent $1/p$ in a fractional Riesz estimate on the hypercube, proposed by Naor, Eskenazis, and Ivanisvili.

September 3, 2026 claimed proof

Zhendong Xu and Hao Zhang claim the estimate for $1<p\le 2$ , including the endpoint, together with higher-order analogues and applications. Their preprint says the result answers the Naor–Eskenazis–Ivanisvili conjecture and uses noncommutative semigroup BMO theory. No referee verification, independent confirmation, error report, or withdrawal was found.

Current status (as of September 2026): The sharp exponent is claimed to be proved for $1<p\le 2$ , but the claim remains unverified and the problem is not independently settled.

Sources

No solutions have been posted yet.