Let 1≤ p≤ q<∞ and z∈C. Weissler's conjecture asserts that the complex noise operator T_z on the Hamming cube has dimension-free norm from L^p to L^q if and only if | p-2-z^2(q-2)|≤ p-q| z|^2. Equivalently, under this condition, one must prove the two-point inequality (frac| a+zb|^q+| a-zb|^q2)^1/q≤(frac| a+b|^p+| a-b|^p2)^1/p for every a,b∈C. The remaining strict cases are 2<p<q<3 and, by duality, 3/2<p<q<2.

Weissler’s missing two-point inequality

Weissler’s conjecture characterizes when the complex two-point inequality holds. The cases left open in the classical account were the strict off-diagonal ranges $2<p<q<3$ and $\frac{3}{2}<p<q<2$ .

Known results

Bonami (1970), Beckner (1975), and Weissler (1979) proved all classical cases except $2<p\leq q<3$ and $\frac{3}{2}<p\leq q<2$ .
Ivanisvili–Nazarov established the diagonal cases $p=q$ in the remaining ranges.

July 2026 proof of the missing cases

The preprint The missing two-point inequality in Weissler’s conjecture proves the two-point inequality for every $2<p<q<\infty$ and $z\in\Omega_{p,q}$ , and obtains the corresponding range $1<p<q<2$ by duality. This includes all previously missing strict cases; tensorization then yields dimension-free complex hypercontractivity on the Hamming cube.

Current status (as of July 2026): The arXiv preprint supplies a proof of all previously missing strict cases, so the problem is resolved subject to normal scholarly verification.

Weissler’s missing two-point inequality

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