On the F-stability and related conjectures

Given a convex function 0 : [0, 1] ? R and the mean ? f (X) = a ? [0, 1], which Boolean function f maximizes the 0-stability E[F (T-? f(X))] of f ? Here X is a random vector uniformly distributed on the discrete cube {-1, 1}n and T(? )is the Bonami-Beckner operator. Special cases of this problem include the (symmetric and asymmetric) a-stability problems and the "Most Informative Boolean Function" problem. In this paper, we provide several upper bounds for the maximal 0-stability. When specializing 0 to some particular forms, by these upper bounds, we partially resolve Mossel and O'Donnell's conjecture on a-stability with a > 2, Li and Medard's conjecture on a-stability with 1 < a < 2, and Courtade and Kumar's conjecture on the "Most Informative Boolean Function" which corresponds to a conjecture on a-stability with a = 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.