Boolean functions: noise stability, non-interactive correlation distillation, and mutual information

Let $T_{epsilon}$ be the noise operator acting on Boolean functions $f:{0, 1}^nto {0, 1}$, where $epsilonin[0, 1/2]$ is the noise parameter. Given $alpha>1$ and fixed mean $mathbb{E} f$, which Boolean function $f$ has the largest $alpha$-th moment $mathbb{E}(T_epsilon f)^alpha$? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumars conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise ($epsilon=epsilon(n)$ is close to 0), high noise ($epsilon=epsilon(n)$ is close to 1/2), as well as when $alpha=alpha(n)$ is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus $(mathbb{Z}/pmathbb{Z})^n$ and the problem of noise stability in a tree model.