On the $Φ$-Stability and Related Conjectures

Let $\mathbf{X}$ be a random variable uniformly distributed on the discrete cube $\left\{ -1,1\right\} ^{n}$, and let $T_{\rho}$ be the noise operator acting on Boolean functions $f:\left\{ -1,1\right\} ^{n}\to\left\{ 0,1\right\} $, where $\rho\in[0,1]$ is the noise parameter, representing the correlation coefficient between each coordination of $\mathbf{X}$ and its noise-corrupted version. Given a convex function $\Phi$ and the mean $\mathbb{E}f(\mathbf{X})=a\in[0,1]$, which Boolean function $f$ maximizes the $\Phi$-stability $\mathbb{E}\left[\Phi\left(T_{\rho}f(\mathbf{X})\right)\right]$ of $f$? Special cases of this problem include the (symmetric and asymmetric) $\alpha$-stability problems and the "Most Informative Boolean Function" problem. In this paper, we provide several upper bounds for the maximal $\Phi$-stability. Considering specific $\Phi$'s, we partially resolve Mossel and O'Donnell's conjecture on $\alpha$-stability with $\alpha>2$, Li and M\'edard's conjecture on $\alpha$-stability with $1<\alpha<2$, and Courtade and Kumar's conjecture on the "Most Informative Boolean Function" which corresponds to a conjecture on $\alpha$-stability with $\alpha=1$. Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut--Kalai--Naor (FKN) theorem. Our improvements of the FKN Theorem are sharp or asymptotically sharp for certain cases.