The optimal hypercontractive constants for ${\mathbb{Z}}_3$ and biased Bernoulli random variables

We resolve a folklore problem of determining the optimal hypercontractive constants $r_{p,q}({\mathbb{Z}}_3)$ for the cyclic group ${\mathbb{Z}}_3$ for all $1<p<q<\mathrm{\infty }$. More precisely, we have $$r_{p,q}({\mathbb{Z}}_3)=\frac{(1+2x)(1-y)}{(1+2y)(1-x)},$$ where $(x,y)$ is the unique solution in the open unit square $(0,1)\times (0,1)$ to the system of equations \begin{align*} \left{ \begin{aligned} &\frac{1}{1+2x}\Big(\frac{1+2x^p}{3}\Big)^{\frac{1}{p}}=\frac{1}{1+2y}\Big(\frac{1+2y^q}{3}\Big)^{\frac{1}{q}},\ &\frac{(1-x)(1-x^{p-1})}{1+2x^p}=\frac{(1-y)(1-y^{q-1})}{1+2y^q}. \end{aligned} \right. \end{align*} Consequently, for rational $p,q\in \mathbb{Q}$, the constants $r_{p,q}({\mathbb{Z}}_3)$ are algebraic numbers which generally admit no radical expressions, since their often rather complicated minimal polynomials may have non-solvable Galois groups. Our formalism relies on a key observation: the existence of nontrivial critical extremizers. This approach can also be adapted to resolve a long-standing open problem – determining all optimal $(p,q)$-hypercontractive constants for biased Bernoulli random variables, which are closely related to noise operators. Several noteworthy phenomena emerge from numerical simulations: the monotonicity of the hypercontractive constants in the parameters, and the appearance of intriguing limit shapes. These phenomena merit further investigation.