Let $T_{\epsilon}$ be the noise operator acting on Boolean functions $f:\{0, 1\}^n\to \{0, 1\}$, where $\epsilon\in[0, 1/2]$ 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-Kumar's 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 $(\mathbb{Z}/p\mathbb{Z})^n$ and the problem of noise stability in a tree model.

翻译：$T ⁇ epsilon}$T ⁇ epsilon}$T ⁇ eplean 函数操作的噪音操作员$f:@0, 1 ⁇ n\to ⁇ 0, 1 ⁇ _$, 其中$\epsilon\ in[0, 1/2] 是一个噪音参数。 $alpha> 1$, 固定平均值$mathbb{E} f$, 布林函数的美元是第1秒最大 $\ alphu{E}( T ⁇ psilon f) 美元? 这个问题与布林函数的噪音稳定性、 非交互性相关蒸馏问题和 courtade- Kumar 在信息性最强的布林函数上的猜想有密切联系。 在本文中,我们在一些极端环境中的最大化者,例如低噪音($\epsilon ⁇ epsib} 美元接近0, 高噪音( eepsilon) 接近 1/2) 。 以及当 $\\ alphalphalpha( n) 是大型的离离( brob) robral__br) 问题。