Mendel–Naor heat-smoothing conjecture for tail functions

Let $1<p<\infty$. Let $f\colon\{-1,1\}^{n}\to\mathbb{R}$ satisfy

$$\mathbb{E} f W_{S}=0$$

for every $S\subseteq\{1,\ldots,n\}$ with $|S|<k$, where $P_t$ is the heat semigroup and $L$ is its generator. Mendel–Naor's heat-smoothing conjecture. There exists $c(p)>0$ such that, for every $t>0$,

$$\|P_t f\|_p\leq e^{-tkc(p)}\|f\|_p,$$

and

$$\|Lf\|_p\geq c(p)k\|f\|_p.$$

The first inequality is the conjectured extension of the known higher-order heat-smoothing estimate from $p\geq2$ to all $1<p<\infty$, with linear dependence on $t$; the second is the corresponding higher-order Poincaré inequality. The paper proves these estimates, so the conjecture is resolved.