On the $\Phi$-Stability and Related Conjectures

Given a convex function $\Phi:[0,1]\to\mathbb{R}$ and the mean $\mathbb{E}f(\mathbf{X})=a\in[0,1]$, which Boolean function $f$ maximizes the $\Phi$-stability $\mathbb{E}[\Phi(T_{\rho}f(\mathbf{X}))]$ of $f$? Here $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{-1,1\}^{n}$ and $T_{\rho}$ is the Bonami-Beckner operator. 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. When specializing $\Phi$ to some particular forms, by these upper bounds, 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.