= Hypercontractive weighted influence bound
{title2=$\sum_{S\ne\varnothing}|S|\delta^{|S|-1}\widehat f(S)^2\leq\frac14\sum_i\beta_i^{2/(1+\delta)}$}
For a zero-one <Boolean function>, the flip-probability <influence> is $\beta_i=4\sum_{S\ni i}\widehat f(S)^2$. Apply <Beckner's inequality> with noise parameter $\sqrt\delta$ to the <discrete derivative of a Boolean function>, whose nonzero magnitude is $1/2$. Its $L^{1+\delta}$ norm squared is $\beta_i^{2/(1+\delta)}/4$. Summing over coordinates gives the displayed weighted <Fourier weight> bound, a useful intermediate step in influence lower bounds.
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