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

Given a convex function $\mathrm{\Phi }:[0,1]\to \mathbb{R}$, the $\mathrm{\Phi }$-stability of a Boolean function $f$ is defined as $\mathbb{E}[\mathrm{\Phi }(T_{\rho }f(\mathbf{X}))]$, where $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{\pm 1{\}}^n$ and $T_{\rho }$ is the Bonami-Beckner operator. In this paper, we prove that dictator functions are locally optimal in maximizing the $\mathrm{\Phi }$-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 $\rho \in [0,0.914]$ or for $q\in [1.36,2)$ and all $\rho \in [0,1]$. In other words, we confirm the (balanced) Courtade–Kumar conjecture with the correlation coefficient $\rho \in [0,0.914]$ 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.