Local Optimality of Dictator Functions with Applications to Courtade--Kumar and Li--Médard Conjectures

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.