Given a convex function $Φ:[0,1]\to\mathbb{R}$, the $Φ$-stability of a Boolean function $f$ is defined as $\mathbb{E}[Φ(T_ρf(\mathbf{X}))]$, where $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{\pm1\}^{n}$ and $T_ρ$ is the Bonami-Beckner operator. In this paper, we prove that dictator functions are locally optimal in maximizing the $Φ$-stability of $f$ over all balanced Boolean functions. When focusing on the symmetric $q$-stability, combining this result with our previous bound, we use computer-assisted methods to prove that dictator functions maximize the symmetric $q$-stability for $q=1$ and $ρ\in[0,0.914]$ or for $q\in[1.36,2)$ and all $ρ\in[0,1]$. In other words, we confirm the (balanced) Courtade--Kumar conjecture with the correlation coefficient $ρ\in[0,0.914]$ and the (symmetrized) Li--Médard conjecture with $q\in[1.36,2)$. We conjecture that dictator functions maximize both the symmetric and asymmetric $\frac{1}{2}$-stability over all balanced Boolean functions. Our proofs are based on majorization of noise operators and hypercontractivity inequalities.

翻译：给定凸函数$Φ:[0,1]\to\mathbb{R}$，布尔函数$f$的$Φ$-稳定性定义为$\mathbb{E}[Φ(T_ρf(\mathbf{X}))]$，其中$\mathbf{X}$是均匀分布在离散立方体$\{\pm1\}^{n}$上的随机向量，$T_ρ$为Bonami-Beckner算子。本文证明，在所有平衡布尔函数中，独裁函数在最大化$Φ$-稳定性方面具有局部最优性。当聚焦于对称$q$-稳定性时，结合此结果与先前的界，我们采用计算机辅助方法证明：对于$q=1$且$ρ\in[0,0.914]$，或对于$q\in[1.36,2)$且所有$ρ\in[0,1]$，独裁函数最大化对称$q$-稳定性。换言之，我们以相关系数$ρ\in[0,0.914]$验证了（平衡）Courtade–Kumar猜想，并以$q\in[1.36,2)$验证了（对称化）Li–Médard猜想。我们推测，在所有平衡布尔函数中，独裁函数同时最大化对称和非对称的$\frac{1}{2}$-稳定性。证明基于噪声算子的优势化与超压缩不等式。