The least-atom principle for optimal hypercontractivity of finite probability spaces

Let $\mu$ be a probability measure on a finite set $\Omega$, and let $\lambda\in(0,1/2)$ be the mass of its least atom. For $1<p<q<\infty$, let $\sigma_{p,q}(\mu)$ be the optimal constant for the operator $\mathbb{E}_\mu+r(\operatorname{Id}-\mathbb{E}_\mu)$ from $L^p(\mu)$ to $L^q(\mu)$, and let $\sigma_{p,q}(\lambda)$ denote the corresponding constant for the biased Bernoulli measure of parameter $\lambda$.

Least-atom principle. For all $1<p<q<\infty$,

$$\sigma_{p,q}(\mu)=\sigma_{p,q}(\lambda).$$

This asserts that the optimal constant depends only on the smallest atom of the finite probability space. It is motivated by numerical simulations and the cited result of Wolff, but the supplied text does not establish it for the full range $1<p<q<\infty$.