It proves some nice facts, including:

- If $f : \{-1,1\}^n \to \{-1,1\}$ has $Var[f] \leq p$, then $W_1(f) = \sum_{|S| = 1} \hat{f}(S)^2 \leq O(p \log (1/p))$. (This is a fact that I seem to use all the time in my work...)
- If $\mathbb{I}\mathbb{I}(f)$ denotes $\sum_i Inf_i(f)^2$, then $W_2(f) = \sum_{|S| = 2} \hat{f}(S)^2 \leq O(\mathbb{I}\mathbb{I}(f) \log(1/\mathbb{I}\mathbb{I}(f)))$. (This is the main technical lemma/theorem in the paper.)
- There is a monotone function $f : \{-1,1\}^n \to \{-1,1\}$ for which a constant fraction of all $x$ have the property that $f$ has $\Omega(\sqrt{n})$ sensitive coordinates on $x$. This immediately implies that the function has $\mathbb{I}(f) \geq \Omega(\sqrt{n})$, but in a much different way than $\mathbb{I}(Maj_n) \geq \Omega(\sqrt{n})$. With Majority, most $x$'s have $0$ sensitive coordinates, but a $\Omega(1/\sqrt{n})$ fraction of them have $(n+1)/2$ sensitive coordinates. With this function, the distribution on the number of sensitive coordinates is much simpler -- a constant fraction have $\Omega(\sqrt{n})$. The function is constructed randomly -- a random DNF of width $\sqrt{n}$ and size $(\ln 2) 2^{\sqrt{n}}$.

An important improvement to the second bullet point above came from a notable paper of Benjamini, Kalai, and Schramm: They showed that in fact $W_k(f) \leq O(\mathbb{I}\mathbb{I} (f) \log^{k-1}(1/\mathbb{I}\mathbb{I}(f))$. I'm trying to use this fact right now for a problem I'm working on...

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