Hypercontractive weighted influence bound (${\sum }_{S\ne \varnothing }|S|{\delta }^{|S|-1}\hat{f}(S{)}^2\le \frac{1}{4}{\sum }_i{\beta }_i^{2/(1+\delta )}$)

For a zero-one Boolean function, the flip-probability influence is ${\beta }_i=4{\sum }_{S\ni i}\hat{f}(S{)}^2$. 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.