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The optimal hypercontractive constants for $\mathbb{Z}_3$

Jie Cao, Shilei Fan, Yong Han, Yanqi Qiu, Zipeng Wang

Abstract

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.

The optimal hypercontractive constants for $\mathbb{Z}_3$

Abstract

We resolve a folklore problem for the optimal hypercontractive constant of the cyclic group for all . Precisely, the optimal hypercontractive constant is given by where is the solution in the open unit square 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 , the constants are algebraic numbers whose minimal polynomials are generally rather complicated, with non-solvable Galois groups and therefore no explicit radical expressions.
Paper Structure (34 sections, 30 theorems, 274 equations, 14 figures, 2 tables)

This paper contains 34 sections, 30 theorems, 274 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Outlines for the proof of Theorem \ref{['thm-main-thm-one-xy']}
  4. Reducing the full space to a segment
  5. Establishing the system of equations
  6. Uniqueness of the solution
  7. Preliminaries
  8. Orthogonal basis method
  9. Monotonicity game
  10. Parameter reduction: from $\mathbb{C}^3$ to $[0,1]$
  11. Proof of Lemma \ref{['lem-rough']}
  12. Reduction from $\mathbb{C}^3$ to $\Delta$ via symmetry
  13. Reduction from $\Delta$ to $[0,1]$ via monotonicity
  14. Proof of Lemma \ref{['lemma-reduce-p-q-1']}
  15. Proof of Lemma \ref{['lemma-reduce-p-q-2']}
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $1 < p < q < \infty$. Then the hypercontractive constant for $\mathbb{Z}_3$ is given by where $(x,y)$ is the unique solution in the open unit square $(0,1)\times(0,1)$ to the system of equations Moreover, the function $f=1 + \frac{1-x}{1+2x} (\chi +\overline{\chi})$ is a nontrivial extremizer.

Figures (14)

  • Figure 1: Nonmultiplicative relationship
  • Figure 2: Reduction architecture
  • Figure 3: Defect functions $G(r,\cdot)$ for different $r$
  • Figure 4: Intersection of the curves $h(p,\cdot)$ and $h(q,\cdot)$
  • Figure 5: The family of curves $H(\alpha,\cdot)$ and a close-up view near $(0,0)$
  • ...and 9 more figures

Theorems & Definitions (79)

  • Theorem 1.1
  • Remark
  • Lemma 1.2
  • Corollary 1.3
  • Remark
  • Corollary 1.4
  • Remark
  • Remark
  • Remark
  • Lemma 2.1
  • ...and 69 more