The missing two-point inequality in Weissler's conjecture

Abstract

Weissler's conjectured characterization of complex hypercontractivity on the Hamming cube remained open in the ranges $2<p\le q<3$ and $\frac{3}{2}<p\le q<2$. Ivanisvili--Nazarov established the diagonal cases. We prove the remaining strict off-diagonal cases through the corresponding two-point inequality. In fact, our argument proves the two-point inequality for every $2<p<q<\infty$ and, by duality, for every $1<p<q<2$.