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 $1 < p < q < \infty$. 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.