We resolve a folklore problem for the optimal hypercontractive constant of the cyclic group $\mathbb{Z}_3$ for all $1 < p < q < \infty$. Precisely, the optimal hypercontractive constant is given by \[ r_{p,q}(\mathbb{Z}_3) = \frac{(1 + 2x)(1 - y)}{(1 + 2y)(1 - x)}, \] where $(x,y)$ is the $\textit{unique}$ 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*} One feature of our formalism is that the system simultaneously determines the optimal hypercontractive constant and a nontrivial extremizer. As a consequence of our main result, for rational $p, q\in \mathbb{Q}$, the constants $r_{p,q}(\mathbb{Z}_3)$ are algebraic numbers whose minimal polynomials are generally rather complicated, with non-solvable Galois groups and therefore no explicit radical expressions.
where (x, y) is the unique solution in the open unit square (0, 1) × (0, 1) to the system of equations
(1 -x)(1 -x p-1 ) 1 + 2x p = (1 -y)(1 -y q-1 ) 1 + 2y q .
- Introduction 1.1. Main results. Given an integer n ≥ 2, let Z n = Z/nZ denote the finite cyclic group equipped with the Haar measure m normalized by m(Z n ) = 1. Throughout the paper, let χ ∈ Z n be the character with χ(1) = e i2π/n . For any r ∈ [0, 1] and any p > 1, define the dilation operator on L p (Z n ) = L p (Z n , m) as
where a k ∈ C for all 0 ≤ k ≤ n -1 and d(k) denotes the graph distance given by d(k) = min{k, n -k} for 0 ≤ k ≤ n -1. Let 1 < p < q < ∞. The hypercontractivity problem for Z n aims to determine the optimal (largest) constant r p,q (Z n ) ∈ [0, 1] such that the following hypercontractive inequality holds:
∥T r f ∥ q ≤ ∥f ∥ p for all 0 ≤ r ≤ r p,q (Z n ) and all f ∈ L p (Z n ).
A non-constant function f for which the hypercontractive inequality becomes an equality when r = r p,q (Z n ) is called a non-trivial (p, q)-extremizer for Z n or simply a non-trivial extremizer.
In the case of Z 2 , the hypercontractive inequality is referred to as the two-point inequality in literature and it was proved by Bonami [Bon70]-Nelson [Nel73]-Beckner [Bec75]-Gross [Gro75] that r p,q (Z 2 ) = (p -1)/(q -1).
And, it has been established that the hypercontractive constant coincides with that of Z 2 for many other cases on finite cyclic groups.
• n = 4: r p,q (Z 4 ) = r p,q (Z 2 ) by Beckner, Janson, and Jerison [BJJ83] in 1983.
• n = 5: r 2,2q (Z 5 ) = r 2,2q (Z 2 ) with q ∈ Z + by Andersson [And02] in 2002.
• n ≥ 6: (either n is odd with n ≥ q or n is even), r 2,2q (Z n ) = r 2,2q (Z 2 ) with q ∈ Z + by Junge, Palazuelos, Parcet and Perrin [JPPP17] in 2017.
• n ∈ {3 • 2 k , 2 k } with k ≥ 1, r p,q (Z n ) = r p,q (Z 2 ) by Yao [Yao25] in 2025. However, the hypercontractivity of Z 3 differs drastically from all the established cases and some partial results on r p,q (Z 3 ) were obtained by Andersson [And02], Diaconis and Saloff-Coste [DSC96]. And Wolff proved in [Wol07, Corollary 3.1] that r 2,q (Z 3 ) = 2(4 1/q -1)/(4 -4 1/q ). Wolff’s result is based on Lata la and Oleszkiewicz [LaO00,Ole03], where the condition p = 2 is crucially used. More details and applications of hypercontractive inequalities on discrete space can be found in [And99, CLSC08, O’D14, JPPR15, IN22, KLLM24, SVZ24] and references therein.
By developing powerful combinatorial techniques, Junge, Palazuelos, Parcet, and Perrin [JPPP17] obtained optimal hypercontractive constant for free groups, some Coxeter groups, and many finite cyclic groups. However, in [JPPP17, after Theorem A2 in page x], the authors note: Z 3 is the simplest group with a 3-loop in its Cayley graph, and the hypercontractive constant is not even conjectured. This inspires us to move away from seeking an explicit closed-form expression for r p,q (Z 3 ), which appears unlikely to exist, and instead to analyze its implicit functional structure. Through this shift in perspective, we succeed in determining r p,q (Z 3 ) for all 1 < p < q < ∞, thereby resolving the problem.
Theorem 1.1. Let 1 < p < q < ∞. Then the hypercontractive constant for Z 3 is given by r p,q (Z 3 ) =
(1 + 2x)(1 -y)
(1 + 2y)(1 -x) , (1.1) where (x, y) is the unique solution in the open unit square (0, 1) × (0, 1) to the system of equations
(1 -x)(1 -x p-1 ) 1 + 2x p = (1 -y)(1 -y q-1 ) 1 + 2y q .
(1.2)
Moreover, the function f = 1 + 1-x 1+2x (χ + χ) is a nontrivial extremizer.
In seeking the optimal constant r p,q (Z 3 ), we unexpectedly discover the existence of nontrivial (p, q)-extremizers, which is in sharp contrast to the classical situation. The existence of such non-trivial extremizer seems to be related to the strict inequality r p,q (Z 3 ) < (p -1)/(q -1) and it naturally allows us to obtain an equation-system via the simplest variational principle (that is, the Fermat principle). However, the uniqueness of solution to the corresponding equation-system is highly nontrivial: indeed, inspired by the theory of Whitney pleats [AGV85,§1], we are led to analyze the intersection properties of a curve-family associated to (1.2) by using symmetrization and blowup techniques.
Remark. The formula (1.1) for r p,q (Z 3 ) is exactly the cross ratio (see, e.g., [Bea95,§4.4] for its definition) of x, y, -1/2, 1: r p,q (Z 3 ) = (x, y; -1/2, 1) := (x + 1/2)(y -1) (x -1)(y + 1/2) . (1.3) Currently, the geometric connection between r p,q (Z 3 ) and the cross ratio is not known to us.
It is clear that the hypercontractive constant r p,q (Z 3 ) has a symmetry of duality: r p,q (Z 3 ) = r q * ,p * (Z 3 ), (1.4) where p * = p/(p-1), q * = q/(q-1) are the corresponding conjugate exponents. On the other hand, the equation-system (1.2) also processes a symmetry of duality and admits a self-dual equivalent formulation as follows. For any 1 < p < ∞ and x ∈ (0, 1), define ℓ(p, x) := 1 1 + 2x 1 + 2x p 3 1 p .
Lemma 1.2. The equation-syste
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